Statistics and Data Distributions

Lesson: Formulating Statistical Questions

In statistics, we often ask questions so we can collect data and learn something from it. But not every question is a statistical question.

A statistical question is a question that expects different answers from different people, objects, or times. It is answered by collecting data that can vary.

A non-statistical question has one exact answer, or it does not expect much variation. These questions do not need a set of different data values to answer them.

Learning the difference is important because statistics is about studying variability. Variability means that the data values are not all the same.

1. What makes a question statistical?

A question is statistical if it does both of these things:

• It can be answered by collecting data.
• It expects the data to have variation, meaning the answers will not all be identical.

For example, think about the question: "How many minutes do 6th grade students spend reading each night?"

This is a statistical question because if you ask many 6th grade students, their answers will probably be different. Some may read for 10 minutes, some for 20 minutes, and some for 30 minutes or more.

2. What makes a question non-statistical?

A non-statistical question usually asks for one exact fact.

For example: "How many days are in one week?"

This is not a statistical question because the answer is always 7. There is no variability.

Another example is: "What is the height of the classroom door?"

If you mean one specific door, then the question has one measurement answer. It does not ask for a group of data values that vary.

3. A helpful way to decide

When you read a question, ask yourself:

1. Am I collecting data from a group, or am I just finding one fact?
2. Do I expect the answers to be different?

If the answers are expected to vary, then the question is likely statistical.

If there is only one exact answer, then it is not statistical.

4. Look for words that suggest variability

Statistical questions often include words or ideas like these:

• students, people, cars, pets
• how many, how much, how long
• in our class, in the school, during the week

These clues often show that you are asking about a group, and groups usually have different values.

But be careful: just because a question says how many does not automatically make it statistical.

For example, "How many months are in a year?" is still not statistical, because the answer is always 12.

5. Changing a non-statistical question into a statistical question

Sometimes a question is close to being statistical, but it needs to be rewritten.

Here is a non-statistical question: "What is my shoe size?"

This has one answer for one person. It is not statistical.

Now change it to: "What are the shoe sizes of students in our class?"

Now it is statistical because different students will likely have different shoe sizes.

Here is another example:

• Non-statistical: "How old is the principal?"
• Statistical: "How old are the teachers at our school?"

The second question expects a range of ages, not one exact answer.

6. Worked Examples

Example 1: Is this a statistical question?

Question: "How many letters are in the word 'triangle'?"

Step 1: Does it ask for data from a group? No.

Step 2: Will the answer vary? No.

Answer: This is not a statistical question.

There is one exact answer: the word triangle has 8 letters.

We can write that as $$8$$ letters.

Example 2: Is this a statistical question?

Question: "How many siblings do students in our class have?"

Step 1: Does it ask for data from a group? Yes, it asks about students in the class.

Step 2: Will the answers vary? Yes. Some students may have 0 siblings, some may have 1, 2, or more.

Answer: This is a statistical question.

A possible set of answers might be:

$$0,\ 1,\ 2,\ 2,\ 3,\ 1,\ 4$$

These values show variability because they are not all the same.

Example 3: Rewrite the question

Original question: "What is the temperature outside right now?"

This is not a statistical question because it asks for one measurement at one time.

To make it statistical, we need a question that expects different data values.

Improved question: "What are the outdoor temperatures at noon during the next 10 days?"

Now the answers can vary from day to day, so it is a statistical question.

A possible data set could be:

$$72,\ 74,\ 71,\ 69,\ 73,\ 75,\ 70,\ 68,\ 72,\ 74$$

Example 4: Compare two questions

Question A: "How tall is Maria?"

Question B: "How tall are the girls in Maria's basketball team?"

Question A is non-statistical because it asks about one person and expects one measurement.

Question B is statistical because it asks about a group, and the heights will likely vary.

So only Question B is a statistical question.

7. Common mistakes to avoid

• Mistake 1: Thinking every question with numbers is statistical. A question can include numbers and still have only one exact answer.
• Mistake 2: Thinking every question about a group is statistical. If all answers are guaranteed to be the same, then it is not statistical.
• Mistake 3: Forgetting about variability. Variability is the key idea in statistical questions.

8. Practice thinking

Decide whether each question is statistical or non-statistical:

• "How many books does each student in our class have at home?"
• "How many wheels does a bicycle have?"
• "How long does it take students in our class to get to school?"
• "What is the name of our school?"

Answers:

• Statistical — book counts will vary.
• Non-statistical — a bicycle has one exact number of wheels, which is 2.
• Statistical — travel times will vary.
• Non-statistical — there is one exact school name.

9. Summary

A statistical question is a question that expects variation in the answers. It is answered by collecting data from a group or over time.

A non-statistical question has one exact answer and does not expect variability.

To tell the difference, ask yourself: "Will the answers vary?" If yes, the question is probably statistical.

Measures of Center: Mean, Median, and Mode

When we collect data, we often want to know what a typical value is. For example, if a class records how many books students read, we may want one number that describes the center of the data.

Three common ways to describe the center of a data set are mean, median, and mode. These are called measures of center.

In this lesson, you will learn what each one means, how to find each one, and when each one is helpful.

1. What is a data set?

A data set is a group of numbers collected for a purpose. Here is an example data set showing the number of minutes 6 students read last night:

\(15, 20, 20, 25, 30, 40\)

We can use this data set to find the mean, median, and mode.

2. Mean

The mean is what many people call the average. To find the mean, add all the numbers in the data set and then divide by how many numbers there are.

Steps for finding the mean:

• Add all the values.
• Count how many values there are.
• Divide the total by the number of values.

For the data set \(15, 20, 20, 25, 30, 40\):

$$ 15+20+20+25+30+40=150 $$

There are \(6\) numbers, so:

$$ \text{Mean} = \frac{150}{6} = 25 $$

So, the mean is 25.

You can think of the mean as the number each value would be if the total were shared equally.

3. Median

The median is the middle value in a data set when the numbers are placed in order from least to greatest.

Steps for finding the median:

• Put the numbers in order from least to greatest.
• Find the middle number.
• If there are 2 middle numbers, add them and divide by 2.

Our data set is already in order:

\(15, 20, 20, 25, 30, 40\)

There are \(6\) numbers, which is an even number. That means there are 2 middle numbers: the 3rd and 4th values.

The 3rd and 4th values are \(20\) and \(25\).

$$ \text{Median} = \frac{20+25}{2} = \frac{45}{2} = 22.5 $$

So, the median is 22.5.

4. Mode

The mode is the value that appears most often in a data set.

In the data set \(15, 20, 20, 25, 30, 40\), the number \(20\) appears 2 times. All the other numbers appear only once.

So, the mode is 20.

Important notes about mode:

• A data set can have one mode.
• A data set can have more than one mode if several numbers tie for appearing most often.
• A data set can have no mode if no number repeats.

5. Comparing mean, median, and mode

Each measure of center tells us something a little different.

• Mean: uses every value in the data set.
• Median: shows the middle value when data is ordered.
• Mode: shows the most common value.

Sometimes all three are close together. Sometimes they are different.

If one number in the data set is much larger or much smaller than the others, the mean can change a lot. The median is often less affected by that kind of unusual value.

For example, look at this data set:

\(8, 9, 10, 10, 11, 50\)

Most values are near \(10\), but \(50\) is much larger than the others.

$$ \text{Mean} = \frac{8+9+10+10+11+50}{6} = \frac{98}{6} \approx 16.3 $$

The mean is about \(16.3\), even though most of the data is near \(10\).

The median is the average of the 3rd and 4th values:

$$ \text{Median} = \frac{10+10}{2} = 10 $$

The mode is also \(10\).

This shows that the median and mode may better describe the center when one value is far away from the rest.

6. Worked Examples

Example 1: Find the mean

The data set is:

\(6, 8, 10, 12, 14\)

Step 1: Add the numbers.

$$ 6+8+10+12+14=50 $$

Step 2: Count how many numbers there are. There are \(5\).

Step 3: Divide.

$$ \text{Mean} = \frac{50}{5} = 10 $$

Answer: The mean is 10.

Example 2: Find the median

The data set is:

\(3, 7, 9, 11, 15\)

The numbers are already in order. There are \(5\) numbers, so the median is the middle one.

The middle number is the 3rd value, which is \(9\).

Answer: The median is 9.

Example 3: Find the mode

The data set is:

\(4, 6, 6, 7, 9, 9, 9, 12\)

Count how many times each number appears:

• \(4\): 1 time
• \(6\): 2 times
• \(7\): 1 time
• \(9\): 3 times
• \(12\): 1 time

The number \(9\) appears most often.

Answer: The mode is 9.

Example 4: Find all three measures of center

The data set is:

\(5, 7, 7, 8, 10, 13\)

Mean:

$$ 5+7+7+8+10+13=50 $$ $$ \text{Mean} = \frac{50}{6} \approx 8.3 $$

Median:

There are \(6\) numbers, so find the average of the 3rd and 4th values. Those values are \(7\) and \(8\).

$$ \text{Median} = \frac{7+8}{2} = 7.5 $$

Mode:

The number \(7\) appears twice. All other numbers appear once.

So the mode is \(7\).

Answers:

• Mean: \(8.3\) (about)
• Median: \(7.5\)
• Mode: \(7\)

7. How to decide which measure is most useful

Sometimes one measure of center is more helpful than the others.

• Use the mean when you want to use all the values in the data set.
• Use the median when you want the middle value, especially if one number is much bigger or smaller than the rest.
• Use the mode when you want to know the most common value.

For example:

• If you want the average quiz score, the mean is often useful.
• If you want a typical house price when one house is much more expensive than the others, the median may be better.
• If you want to know the most common shoe size in a class, the mode is helpful.

8. Common mistakes to avoid

• Forgetting to put numbers in order before finding the median.
• Dividing by the wrong number when finding the mean. Always divide by how many values are in the data set.
• Choosing the largest number for the mode. The mode is not the largest number; it is the number that appears most often.
• Thinking every data set has a mode. Some data sets have no repeated values, so they have no mode.

9. Quick check

Find the mean, median, and mode of this data set:

\(2, 4, 4, 6, 8\)

Mean:

$$ 2+4+4+6+8=24 $$ $$ \text{Mean} = \frac{24}{5} = 4.8 $$

Median:

There are \(5\) numbers, so the middle value is the 3rd number: \(4\).

Mode:

The number \(4\) appears most often.

Answers:

• Mean: \(4.8\)
• Median: \(4\)
• Mode: \(4\)

Summary

Mean, median, and mode are all ways to describe the center of a data set.

• The mean is found by adding all values and dividing by the number of values.
• The median is the middle value in an ordered list.
• The mode is the value that appears most often.

By learning all three, you can better understand data and choose the measure of center that best describes what is typical.

Measures of Variability: Range and IQR

When we look at a set of data, we do not only want to know the middle or typical value. We also want to know how spread out the data is. This is called variability.

Two important ways to measure variability are the range and the interquartile range, or IQR. These help us describe whether data values are close together or far apart.

In this lesson, you will learn:

• what range means,
• how to find the range,
• what quartiles are,
• how to find the IQR,
• and how range and IQR help describe data.

1. What is variability?

Variability tells us how much the numbers in a data set differ from each other. If the numbers are all close together, the variability is small. If the numbers are spread far apart, the variability is large.

For example, compare these two data sets:

• Set A: 7, 8, 8, 9, 8
• Set B: 2, 6, 8, 10, 14

Both sets have a middle value near 8, but Set B is much more spread out. That means Set B has greater variability.

2. Range

The range is the difference between the greatest value and the least value in a data set.

The formula is:

$$\text{Range} = \text{greatest value} - \text{least value}$$

To find the range:

1. Put the data in order from least to greatest.
2. Find the smallest number.
3. Find the largest number.
4. Subtract.

Worked Example 1: Finding the Range

A student records the number of books read by 6 friends in one month:

3, 7, 5, 9, 4, 8

First, put the data in order:

3, 4, 5, 7, 8, 9

The least value is 3. The greatest value is 9.

$$\text{Range} = 9 - 3 = 6$$

Answer: The range is 6.

What range tells us: The data values stretch across 6 units from the smallest to the largest.

3. A note about outliers

Sometimes one value is much smaller or much larger than the rest. This kind of value can make the range much bigger. A value like that is often called an outlier.

That is one reason why range is useful, but sometimes it does not tell the whole story. To understand the spread of the middle part of the data, we use the IQR.

4. Quartiles

The word quartile comes from a word meaning “four parts.” Quartiles split ordered data into 4 equal parts.

To find quartiles, the data must be listed from least to greatest.

The three important quartiles are:

• First quartile, or \(Q_1\): the middle of the lower half of the data
• Second quartile: the median of the whole data set
• Third quartile, or \(Q_3\): the middle of the upper half of the data

For IQR, we focus on \(Q_1\) and \(Q_3\).

5. What is the Interquartile Range?

The interquartile range, or IQR, measures the spread of the middle 50% of the data.

The formula is:

$$\text{IQR} = Q_3 - Q_1$$

This means we subtract the first quartile from the third quartile.

The IQR is useful because it ignores the lowest 25% and highest 25% of the data. This means it is less affected by unusually small or large values.

6. How to find the IQR

Follow these steps:

1. Put the data in order from least to greatest.
2. Find the median of the whole data set.
3. Separate the data into a lower half and an upper half.
4. Find the median of the lower half. This is \(Q_1\).
5. Find the median of the upper half. This is \(Q_3\).
6. Subtract: $$\text{IQR} = Q_3 - Q_1$$

When finding \(Q_1\) and \(Q_3\), do not include the median of the whole set in either half if there is one middle number.

Worked Example 2: Finding the IQR with an Odd Number of Values

Find the IQR of this data set:

2, 4, 5, 7, 8, 10, 12

The data is already in order.

Step 1: Find the median.

There are 7 values, so the median is the middle value, which is 7.

Step 2: Split the data into two halves.

Lower half: 2, 4, 5

Upper half: 8, 10, 12

We do not include the median 7 in either half.

Step 3: Find \(Q_1\) and \(Q_3\).

The median of 2, 4, 5 is 4, so \(Q_1 = 4\).

The median of 8, 10, 12 is 10, so \(Q_3 = 10\).

Step 4: Find the IQR.

$$\text{IQR} = Q_3 - Q_1 = 10 - 4 = 6$$

Answer: The IQR is 6.

What this means: The middle 50% of the data is spread across 6 units.

Worked Example 3: Finding the IQR with an Even Number of Values

Find the IQR of this data set:

1, 3, 4, 6, 8, 9, 11, 15

The data is already in order.

Step 1: Find the median.

There are 8 values, so the median is the average of the two middle values, 6 and 8.

$$\text{Median} = \frac{6+8}{2} = 7$$

Step 2: Split the data into two halves.

Lower half: 1, 3, 4, 6

Upper half: 8, 9, 11, 15

Step 3: Find \(Q_1\) and \(Q_3\).

For the lower half, the middle two numbers are 3 and 4.

$$Q_1 = \frac{3+4}{2} = 3.5$$

For the upper half, the middle two numbers are 9 and 11.

$$Q_3 = \frac{9+11}{2} = 10$$

Step 4: Find the IQR.

$$\text{IQR} = 10 - 3.5 = 6.5$$

Answer: The IQR is 6.5.

7. Comparing range and IQR

Both range and IQR measure spread, but they do it in different ways.

• Range uses only the smallest and largest values.
• IQR uses the middle 50% of the data.

This means:

• Range can change a lot if there is one unusual value.
• IQR usually gives a better picture of the spread of the middle of the data.

Look at this example:

• Data Set A: 4, 5, 6, 7, 8
• Data Set B: 4, 5, 6, 7, 20

For Data Set A:

$$\text{Range} = 8 - 4 = 4$$

For Data Set B:

$$\text{Range} = 20 - 4 = 16$$

The range became much larger because of the 20.

But the middle values in both sets are still similar. That is why IQR can be helpful when a data set has an unusual value.

Worked Example 4: Finding Both Range and IQR

The daily temperatures for 9 days are:

12, 14, 15, 15, 16, 18, 19, 22, 30

The data is already in order.

Find the range.

Least value = 12

Greatest value = 30

$$\text{Range} = 30 - 12 = 18$$

Find the IQR.

There are 9 values, so the median is the middle value: 16.

Lower half: 12, 14, 15, 15

Upper half: 18, 19, 22, 30

Find \(Q_1\):

The middle two values in the lower half are 14 and 15.

$$Q_1 = \frac{14+15}{2} = 14.5$$

Find \(Q_3\):

The middle two values in the upper half are 19 and 22.

$$Q_3 = \frac{19+22}{2} = 20.5$$

Now subtract:

$$\text{IQR} = 20.5 - 14.5 = 6$$

Answer:

• Range = 18
• IQR = 6

What this tells us: The full data set stretches 18 units, but the middle 50% is packed into just 6 units. The value 30 makes the range larger.

8. Common mistakes to avoid

• Do not forget to order the data first. Range and IQR are much easier to find when the numbers are in order.
• Do not subtract in the wrong direction. Always do greatest minus least for range, and \(Q_3 - Q_1\) for IQR.
• Do not include the median in both halves when there is one middle value.
• Be careful when there are an even number of values. Sometimes you need to average two middle numbers.

9. Quick step-by-step guide

To find range:

1. Order the data.
2. Find the smallest and largest values.
3. Subtract: $$\text{largest} - \text{smallest}$$

To find IQR:

1. Order the data.
2. Find the median.
3. Find the lower half and upper half.
4. Find \(Q_1\), the median of the lower half.
5. Find \(Q_3\), the median of the upper half.
6. Subtract: $$Q_3 - Q_1$$

10. Summary

Range and IQR are both measures of variability. They help us describe how spread out a data set is.

The range uses the smallest and largest values:

$$\text{Range} = \text{greatest value} - \text{least value}$$

The IQR uses the middle 50% of the data:

$$\text{IQR} = Q_3 - Q_1$$

Range is quick to find, but it can be affected a lot by one unusual value. IQR gives a better look at the spread of the middle part of the data.

When you work with data, ask yourself two questions:

• How far apart are the smallest and largest values?
• How spread out is the middle 50%?

If you can answer both, you understand two important ways to describe variability.

Mean Absolute Deviation (MAD) helps us understand how spread out a set of data is.

The mean is the average of the data. The mean absolute deviation, or MAD, tells us how far the data values are from the mean on average.

If the MAD is small, the data values are close to the mean. If the MAD is large, the data values are more spread out.

Think of MAD as a way to measure variability. Variability means how much the numbers in a data set differ from one another.

What does “absolute deviation” mean?

A deviation is the distance between a data value and the mean. We find it by subtracting:

\( \text{deviation} = \text{data value} - \text{mean} \)

But some answers may be negative. Since distance should not be negative, we use the absolute value. Absolute value means how far a number is from 0, without caring about direction.

So the absolute deviation is:

\( \text{absolute deviation} = |\text{data value} - \text{mean}| \)

Then we average all those absolute deviations. That average is the MAD.

Steps for finding MAD

1. Find the mean of the data set.
2. Find the absolute deviation for each data value by subtracting the mean and taking the absolute value.
3. Add all the absolute deviations.
4. Divide by the number of data values.

In a formula, MAD is:

$$ \text{MAD} = \frac{\text{sum of absolute deviations}}{\text{number of data values}} $$

Let’s practice with some examples.

Worked Example 1: A small data set

Find the MAD of the data set: \(4, 6, 8\)

Step 1: Find the mean.

$$ \text{Mean} = \frac{4+6+8}{3} = \frac{18}{3} = 6 $$

Step 2: Find each absolute deviation from the mean.

• For 4: \(|4-6| = 2\)
• For 6: \(|6-6| = 0\)
• For 8: \(|8-6| = 2\)

Step 3: Add the absolute deviations.

$$ 2+0+2=4 $$

Step 4: Divide by the number of values.

$$ \text{MAD} = \frac{4}{3} = 1\frac{1}{3} $$

Answer: The MAD is \(1\frac{1}{3}\).

This means the data values are, on average, \(1\frac{1}{3}\) units away from the mean.

Worked Example 2: Data with repeated values

Find the MAD of the data set: \(2, 2, 4, 6, 6\)

Step 1: Find the mean.

$$ \text{Mean} = \frac{2+2+4+6+6}{5} = \frac{20}{5} = 4 $$

Step 2: Find the absolute deviations.

• \(|2-4|=2\)
• \(|2-4|=2\)
• \(|4-4|=0\)
• \(|6-4|=2\)
• \(|6-4|=2\)

Step 3: Add them.

$$ 2+2+0+2+2=8 $$

Step 4: Divide by 5.

$$ \text{MAD} = \frac{8}{5} = 1.6 $$

Answer: The MAD is \(1.6\).

Worked Example 3: Comparing two data sets

Which data set has more variability?

• Set A: \(5, 5, 5, 5, 5\)
• Set B: \(1, 3, 5, 7, 9\)

We compare variability by comparing the MADs.

Set A

Mean:

$$ \frac{5+5+5+5+5}{5}=5 $$

Absolute deviations:

• \(|5-5|=0\)
• \(|5-5|=0\)
• \(|5-5|=0\)
• \(|5-5|=0\)
• \(|5-5|=0\)

MAD:

$$ \frac{0+0+0+0+0}{5}=0 $$

Set B

Mean:

$$ \frac{1+3+5+7+9}{5}=\frac{25}{5}=5 $$

Absolute deviations:

• \(|1-5|=4\)
• \(|3-5|=2\)
• \(|5-5|=0\)
• \(|7-5|=2\)
• \(|9-5|=4\)

MAD:

$$ \frac{4+2+0+2+4}{5}=\frac{12}{5}=2.4 $$

Answer: Set B has the greater MAD, so Set B has more variability.

This makes sense. In Set A, all values are the same. In Set B, the values are spread out from the mean.

Worked Example 4: A data set with a decimal MAD

Find the MAD of the data set: \(3, 7, 7, 11\)

Step 1: Find the mean.

$$ \text{Mean} = \frac{3+7+7+11}{4} = \frac{28}{4} = 7 $$

Step 2: Find the absolute deviations.

• \(|3-7|=4\)
• \(|7-7|=0\)
• \(|7-7|=0\)
• \(|11-7|=4\)

Step 3: Add them.

$$ 4+0+0+4=8 $$

Step 4: Divide by 4.

$$ \text{MAD} = \frac{8}{4} = 2 $$

Answer: The MAD is \(2\).

Important ideas to remember

• The MAD is always 0 or greater. It can never be negative.
• If all values are the same, the MAD is 0.
• A larger MAD means the data are more spread out.
• A smaller MAD means the data are closer to the mean.

Common mistakes to avoid

• Forgetting to find the mean first. MAD is based on distance from the mean.
• Not using absolute value. Negative and positive deviations should not cancel each other out.
• Dividing by the wrong number. Divide by the number of data values, not by the sum.

Quick check

Try this on your own: Find the MAD of \(1, 2, 3, 4, 5\).

Solution:

Mean:

$$ \frac{1+2+3+4+5}{5}=3 $$

Absolute deviations:

• \(|1-3|=2\)
• \(|2-3|=1\)
• \(|3-3|=0\)
• \(|4-3|=1\)
• \(|5-3|=2\)

Add:

$$ 2+1+0+1+2=6 $$

Divide by 5:

$$ \text{MAD} = \frac{6}{5} = 1.2 $$

Answer: The MAD is \(1.2\).

Summary

Mean Absolute Deviation, or MAD, measures the average distance between each data value and the mean.

To find it, first find the mean. Then find the absolute deviation of each value, add those distances, and divide by the number of values.

MAD helps us describe how much a data set varies. A small MAD means the data are clustered near the mean, and a large MAD means the data are more spread out.

Creating and Interpreting Histograms

When we collect numerical data, we often want to see how the values are spread out. A histogram is a graph that helps us do that.

A histogram shows how many data values fall within certain number ranges. These number ranges are called intervals or bins.

For example, if we recorded the number of minutes students read each night, we might group the data into bins like 0–9 minutes, 10–19 minutes, 20–29 minutes, and so on. Then we count how many students fall into each bin.

This lesson will help you learn how to make a histogram and how to read what it tells you.

What Is a Histogram?

A histogram is a graph used for numerical data that has been grouped into intervals.

• The horizontal axis shows the intervals or bins.
• The vertical axis shows the frequency, which means how many data values are in each bin.
• The bars touch each other because the intervals are connected number ranges.

Important idea: Histograms are different from bar graphs.

• A bar graph is usually used for categories, like favorite fruits or pet types.
• A histogram is used for numerical data grouped into intervals, like heights, test scores, or times.
• In a bar graph, the bars usually do not touch.
• In a histogram, the bars usually do touch.

Steps for Creating a Histogram

1. Collect the data.
2. Choose equal-width bins. This means each interval covers the same amount.
3. Tally the data. Count how many values go into each bin.
4. Draw the axes.
5. Label the horizontal axis with the intervals.
6. Label the vertical axis with frequency.
7. Draw bars for each interval up to the correct frequency.

Choosing Good Bins

The bins should make the data easy to understand. If the bins are too wide, you lose detail. If the bins are too narrow, the graph can be hard to read.

It is also important that the bins are equal in size. For example, these are equal-width bins:

0–4, 5–9, 10–14, 15–19

Each bin covers 5 numbers.

This set would not be a good choice:

0–4, 5–14, 15–19

These bins are not all the same width.

Worked Example 1: Making a Histogram from Data

The number of books read by 15 students over the summer is:

2, 5, 7, 8, 4, 6, 11, 13, 9, 10, 3, 5, 12, 7, 6

Step 1: Choose bins.

We can use these intervals:

• 0–3
• 4–7
• 8–11
• 12–15

Each bin has width 4.

Step 2: Count the frequencies.

• 0–3: 2, 3 → frequency 2
• 4–7: 4, 5, 5, 6, 6, 7, 7 → frequency 7
• 8–11: 8, 9, 10, 11 → frequency 4
• 12–15: 12, 13 → frequency 2

We can organize this in a table:

0–3: 2
4–7: 7
8–11: 4
12–15: 2

Step 3: Draw the histogram.

The horizontal axis shows the bins. The vertical axis shows frequency. The bar heights are 2, 7, 4, and 2.

What does the histogram show?

• Most students read between 4 and 7 books.
• Very few students read fewer than 4 books.
• Very few students read 12 or more books.

Worked Example 2: Reading a Histogram

A class measured the lengths of 20 leaves in centimeters. A histogram was made with these frequencies:

• 0–2 cm: 1
• 3–5 cm: 5
• 6–8 cm: 8
• 9–11 cm: 4
• 12–14 cm: 2

Question 1: Which interval has the greatest frequency?

The interval 6–8 cm has frequency 8, which is the greatest.

Question 2: How many leaves were shorter than 6 cm?

Shorter than 6 cm means the first two bins:

$$1 + 5 = 6$$

So, 6 leaves were shorter than 6 cm.

Question 3: About how many leaves were at least 9 cm long?

At least 9 cm means the last two bins:

$$4 + 2 = 6$$

So, 6 leaves were at least 9 cm long.

Question 4: What does the histogram tell us?

• Most leaves are between 6 and 8 cm.
• Very short leaves and very long leaves are less common.

Worked Example 3: Creating a Histogram with Larger Numbers

Here are the scores on a quiz for 18 students:

54, 61, 67, 72, 75, 77, 81, 83, 84, 86, 88, 90, 91, 73, 69, 64, 79, 85

Step 1: Choose equal-width bins.

Let’s use bins of width 10:

• 50–59
• 60–69
• 70–79
• 80–89
• 90–99

Step 2: Count the scores in each bin.

• 50–59: 54 → 1
• 60–69: 61, 64, 67, 69 → 4
• 70–79: 72, 73, 75, 77, 79 → 5
• 80–89: 81, 83, 84, 85, 86, 88 → 6
• 90–99: 90, 91 → 2

Step 3: Interpret the histogram.

• The tallest bar would be 80–89.
• Most students scored between 80 and 89.
• Only 1 student scored in the 50s.
• 2 students scored in the 90s.

This histogram shows that many scores are in the middle to high range.

How Histograms Help Us Understand Data

Histograms help us answer questions like these:

• Where are most of the data values?
• Which intervals have the highest or lowest frequencies?
• Is the data spread out or clustered together?
• Are there only a few very small or very large values?

When you look at a histogram, pay attention to:

• The tallest bars — these show the most common intervals.
• The shortest bars — these show less common intervals.
• The overall spread — this shows how far the data goes from smallest to largest.

Common Mistakes to Avoid

• Using unequal bin widths. Keep the intervals the same size.
• Leaving spaces between bars. In a histogram, the bars touch.
• Mixing up a histogram and a bar graph. Histograms are for numerical intervals, not categories.
• Counting data into the wrong bin. Check each value carefully.
• Forgetting labels. Always label both axes.

Quick Check

Suppose these are the ages of pets at a vet clinic:

1, 2, 2, 3, 4, 4, 5, 6, 7, 7, 8, 9

Use bins:

• 0–2
• 3–5
• 6–8
• 9–11

Count the frequencies:

• 0–2: 1, 2, 2 → 3
• 3–5: 3, 4, 4, 5 → 4
• 6–8: 6, 7, 7, 8 → 4
• 9–11: 9 → 1

If you drew the histogram, the bar heights would be 3, 4, 4, and 1.

What can we say?

• Most pets are between 3 and 8 years old.
• Very few pets are between 9 and 11 years old.

Summary

A histogram is a graph that shows how numerical data is grouped into intervals. The bars touch because the intervals are connected. To create a histogram, choose equal-width bins, count how many values fall in each bin, and draw bars to show the frequencies.

To interpret a histogram, look for the tallest and shortest bars and notice how the data is spread out. Remember: a histogram is different from a bar graph because it shows grouped numerical data, not separate categories.

Creating and Interpreting Box Plots

A box plot is a graph that helps us understand a set of numerical data. It shows how the data is spread out by using five important values called the five-number summary.

The five-number summary is:

• Minimum: the smallest value
• First quartile (Q1): the middle of the lower half of the data
• Median: the middle value of the whole data set
• Third quartile (Q3): the middle of the upper half of the data
• Maximum: the greatest value

Box plots are useful because they help us quickly see:

• the center of the data
• how spread out the data is
• where the data is clustered
• whether one half of the data is more spread out than another

Step 1: Put the data in order

Before making a box plot, always write the data from least to greatest. This is very important, because the median and quartiles depend on the order.

For example, if the data is:

\(12, 7, 18, 10, 15\)

First put it in order:

\(7, 10, 12, 15, 18\)

Step 2: Find the median

The median is the middle number in an ordered data set.

If there is an odd number of data values, the median is the middle number.

If there is an even number of data values, the median is the average of the two middle numbers.

Example:

• For \(3, 5, 7\), the median is \(5\).
• For \(4, 6, 8, 10\), the median is $$\frac{6+8}{2}=7$$

Step 3: Find Q1 and Q3

After finding the median, split the data into two halves.

• The lower half is the data below the median.
• The upper half is the data above the median.

Then find the median of each half:

• The median of the lower half is Q1.
• The median of the upper half is Q3.

When the data set has an odd number of values, do not include the overall median in either half.

Step 4: Find the minimum and maximum

These are the smallest and largest numbers in the ordered data set.

Step 5: Draw the box plot

To draw a box plot:

1. Draw a number line that fits the data.
2. Mark the minimum, Q1, median, Q3, and maximum.
3. Draw a box from Q1 to Q3.
4. Draw a line inside the box at the median.
5. Draw a line, called a whisker, from the minimum to Q1.
6. Draw another whisker from Q3 to the maximum.

The box shows the middle half of the data. The whiskers show the lower and upper ends of the data.

Worked Example 1: Finding the five-number summary

Data: \(4, 7, 9, 10, 12, 15, 18\)

The data is already in order.

There are 7 values, so the median is the middle value:

Median = \(10\)

Lower half: \(4, 7, 9\)

Upper half: \(12, 15, 18\)

The middle of the lower half is \(7\), so:

\(Q1 = 7\)

The middle of the upper half is \(15\), so:

\(Q3 = 15\)

The minimum is \(4\) and the maximum is \(18\).

So the five-number summary is:

Minimum \(=4\), Q1 \(=7\), Median \(=10\), Q3 \(=15\), Maximum \(=18\)

Worked Example 2: A data set with an even number of values

Data: \(2, 5, 6, 8, 9, 11, 13, 20\)

There are 8 values, so the median is the average of the two middle numbers:

$$\frac{8+9}{2}=8.5$$

Lower half: \(2, 5, 6, 8\)

Upper half: \(9, 11, 13, 20\)

Find Q1 by averaging the middle two numbers in the lower half:

$$Q1=\frac{5+6}{2}=5.5$$

Find Q3 by averaging the middle two numbers in the upper half:

$$Q3=\frac{11+13}{2}=12$$

Minimum \(=2\), Maximum \(=20\)

So the five-number summary is:

Minimum \(=2\), Q1 \(=5.5\), Median \(=8.5\), Q3 \(=12\), Maximum \(=20\)

To draw the box plot:

• Place points at \(2, 5.5, 8.5, 12, 20\) on a number line.
• Draw the box from \(5.5\) to \(12\).
• Draw a line at \(8.5\) inside the box.
• Draw whiskers from \(2\) to \(5.5\) and from \(12\) to \(20\).

How to interpret a box plot

Once a box plot is drawn, you can learn a lot from it.

• The median shows the middle of the data.
• The box shows the middle 50% of the data.
• A longer section means the data in that part is more spread out.
• A shorter section means the data in that part is more closely packed together.

Each part of the box plot contains about one-fourth of the data:

• from minimum to Q1
• from Q1 to median
• from median to Q3
• from Q3 to maximum

If one of these parts is long, then that fourth of the data is spread out. If one part is short, that fourth of the data is crowded close together.

Worked Example 3: Interpreting a box plot

A box plot has these values:

Minimum \(=3\), Q1 \(=6\), Median \(=8\), Q3 \(=14\), Maximum \(=16\)

What can we say about the data?

• The middle value is \(8\), so about half the data is below \(8\) and about half is above \(8\).
• The middle 50% of the data is between \(6\) and \(14\).
• The section from \(8\) to \(14\) is longer than the section from \(6\) to \(8\), so that part of the data is more spread out.
• The section from \(14\) to \(16\) is short, so the highest values are close together.

Worked Example 4: Create and interpret a box plot from data

Data: \(1, 3, 4, 6, 7, 8, 10, 12, 13\)

The data is already in order.

There are 9 values, so the median is the 5th value:

Median \(=7\)

Lower half: \(1, 3, 4, 6\)

Upper half: \(8, 10, 12, 13\)

Find Q1:

$$Q1=\frac{3+4}{2}=3.5$$

Find Q3:

$$Q3=\frac{10+12}{2}=11$$

Minimum \(=1\), Maximum \(=13\)

Five-number summary:

\(1, 3.5, 7, 11, 13\)

Now interpret it:

• The middle 50% of the data is from \(3.5\) to \(11\).
• The lower whisker goes from \(1\) to \(3.5\), which is length \(2.5\).
• The upper whisker goes from \(11\) to \(13\), which is length \(2\).
• The section from \(7\) to \(11\) is longer than the section from \(3.5\) to \(7\), so the data above the median is more spread out than the data just below the median.

Common mistakes to avoid

• Do not forget to order the data first.
• Do not confuse the median with Q1 or Q3. The median is for the whole data set. Q1 and Q3 are medians of the halves.
• When there is an odd number of data values, do not include the median in the lower or upper half.
• Make sure the box starts at Q1 and ends at Q3.
• Make sure the whiskers go to the minimum and maximum.

Helpful way to remember

A box plot is built from 5 numbers:

minimum, Q1, median, Q3, maximum

You can think of it like this:

• Find the smallest and largest values.
• Find the middle of the whole set.
• Find the middle of the lower half.
• Find the middle of the upper half.
• Plot all 5 numbers on a number line.

Summary

A box plot is a simple way to show how data is spread out. To create one, first order the data, then find the minimum, Q1, median, Q3, and maximum. Draw a box from Q1 to Q3, place a line at the median, and draw whiskers to the minimum and maximum.

To interpret a box plot, look at the lengths of the parts. Longer parts mean the data is more spread out there, and shorter parts mean the data is more packed together. Box plots help us compare distributions and understand where values are clustered.

Analyzing the Shape of Data Distributions

When we collect data, we often organize it in a graph such as a dot plot, bar graph, or histogram. Looking at the graph can tell us more than just the smallest and largest values. It can also show us the shape of the data.

The shape of a data distribution is the overall pattern the data makes on a graph. Learning to describe the shape helps us understand what the data is doing.

In this lesson, you will learn how to describe a data distribution as:

• symmetrical,
• skewed left,
• skewed right,
• or having clusters and gaps.

You do not need to do difficult calculations for this skill. You mostly need to look carefully at the graph and describe what you see.

1. What is a data distribution?

A data distribution is the way data values are spread out. If you make a graph of the data, the distribution is the shape formed by the points or bars.

For example, imagine the number of books read by students in one month. If most students read about 4 or 5 books, the graph will be tallest around 4 and 5. If only a few students read 0 or 10 books, the graph will be lower at the ends.

When you study the shape, ask yourself these questions:

• Where are most of the data values?
• Does one side stretch out farther than the other?
• Are there spaces where no data appears?
• Are there groups of values close together?

2. Symmetrical distributions

A distribution is symmetrical when the left side and the right side look about the same. It is like the graph could be folded in half, and both sides would almost match.

In a symmetrical distribution:

• the middle has the most data,
• the ends have less data,
• and both sides are balanced.

Here is a simple example of counts that are symmetrical:

1 value at 1, 2 values at 2, 3 values at 3, 2 values at 4, 1 value at 5

The counts go up and then back down in the same way. That makes the shape symmetrical.

3. Skewed right distributions

A distribution is skewed right when most of the data is on the left or middle, but the graph stretches out farther to the right.

This long stretched part is sometimes called a tail. In a skewed right distribution, the tail points right.

In a skewed right distribution:

• many values are small or medium,
• only a few values are very large,
• and the graph trails off to the right.

For example, if most students finish a quiz in 10 to 15 minutes, but a few students take 20, 25, or 30 minutes, the graph may be skewed right.

4. Skewed left distributions

A distribution is skewed left when most of the data is on the right or middle, but the graph stretches out farther to the left.

In a skewed left distribution, the tail points left.

In a skewed left distribution:

• many values are large or medium,
• only a few values are very small,
• and the graph trails off to the left.

For example, if many students score between 80 and 100 on a test, but only a few score much lower, the graph may be skewed left.

5. Clusters and gaps

Sometimes a graph has parts that are grouped together and parts that are empty.

A cluster is a group of data values that are close together.

A gap is a part of the graph where there are few or no data values.

For example, imagine these values:

2, 3, 3, 4, 4, 15, 16, 16, 17

There is one cluster around 2 to 4 and another cluster around 15 to 17. There is a large gap between 5 and 14 because there are no values there.

Clusters and gaps help us notice that the data may not all belong to one big group.

6. Uniform distributions

Sometimes the graph looks fairly flat, with about the same amount of data in each part. This is called uniform.

In a uniform distribution:

• the bars or dots are spread out evenly,
• there is no big peak in the middle,
• and no side is much longer than the other.

For example, if the numbers 1 through 6 each appear about the same number of times, the graph may look uniform.

7. How to describe the shape of a graph

When you look at a data graph, follow these steps:

1. Find where most of the data is. Look for the tallest bars or the greatest number of dots.
2. Check both sides. Does the graph look balanced, or does one side stretch out more?
3. Look for a tail. A longer tail on the right means skewed right. A longer tail on the left means skewed left.
4. Look for groups and spaces. Groups are clusters. Empty spaces are gaps.
5. Decide whether the graph is symmetrical, skewed, uniform, or has clusters and gaps.

8. Important tip about skew

Students sometimes think the skew direction is where most of the data is. That is not correct.

The direction of the skew is named by the direction of the tail.

• Tail to the right \(\rightarrow\) skewed right
• Tail to the left \(\rightarrow\) skewed left

So even if most data is on the left, the graph can still be skewed right if the long tail goes to the right.

Worked Example 1: Identifying a symmetrical distribution

A dot plot has these counts:

• 1 dot at 2
• 2 dots at 3
• 3 dots at 4
• 2 dots at 5
• 1 dot at 6

Question: What is the shape of the distribution?

Step 1: Look at the center. The most dots are at 4.

Step 2: Compare the sides. The counts on the left of 4 are 2 and 1. The counts on the right of 4 are also 2 and 1.

Step 3: The graph rises to the middle and falls the same way on both sides.

Answer: The distribution is symmetrical.

Worked Example 2: Identifying skewed right

A graph shows the number of minutes students spend reading each night. Most students are between 10 and 20 minutes. A few students are at 25, 30, and 40 minutes.

Question: What is the shape of the distribution?

Step 1: Most of the data is grouped between 10 and 20.

Step 2: The graph stretches farther on the right because of 25, 30, and 40.

Step 3: The tail points right.

Answer: The distribution is skewed right.

Worked Example 3: Identifying skewed left

Test scores for a class are mostly between 75 and 100. Only a few scores are 40, 50, and 60.

Question: What is the shape of the distribution?

Step 1: Most of the scores are on the high end, between 75 and 100.

Step 2: The graph stretches farther on the left because of 40, 50, and 60.

Step 3: The tail points left.

Answer: The distribution is skewed left.

Worked Example 4: Identifying clusters and gaps

Suppose the data values are:

$$2,\ 3,\ 3,\ 4,\ 5,\ 12,\ 13,\ 13,\ 14,\ 20$$

Question: What features does this distribution have?

Step 1: Look for groups of values. There is a group from 2 to 5 and another group from 12 to 14.

Step 2: Look for empty spaces. There are no values from 6 to 11, so that is a gap.

Step 3: Notice that 20 stands alone and is far from the others.

Answer: The distribution has clusters at 2 to 5 and 12 to 14, and a gap from 6 to 11.

9. Comparing different shapes

Here is a quick way to tell the shapes apart:

• Symmetrical: both sides look about the same.
• Skewed right: the tail stretches to the right.
• Skewed left: the tail stretches to the left.
• Uniform: values are spread fairly evenly.
• Clusters: values form groups.
• Gaps: spaces with few or no values.

10. Common mistakes to avoid

• Do not name the skew by where most of the data is. Name it by the direction of the tail.
• Do not assume every graph has only one feature. A graph can be skewed and also have a gap or a cluster.
• Do not focus on just one value. Look at the whole graph.
• Do not worry if a graph is not perfect. Real data often only looks approximately symmetrical or approximately uniform.

11. Practice thinking questions

• If the left and right sides match closely, the graph is probably symmetrical.
• If most values are low and a few large values stretch right, the graph is probably skewed right.
• If most values are high and a few small values stretch left, the graph is probably skewed left.
• If values are grouped in sections, the graph has clusters.
• If there are empty spaces between values, the graph has gaps.

Summary

Analyzing the shape of a data distribution means describing the overall pattern of a graph. A distribution can be symmetrical, skewed left, skewed right, or uniform. It can also have clusters and gaps.

To describe the shape, look for where most of the data is, compare the left and right sides, check for a tail, and notice any groups or empty spaces. The more you practice looking at whole graphs, the easier it becomes to describe data distributions clearly and correctly.

Identifying Outliers and Their Effects

When we collect data, most values often stay close together. But sometimes one value is much bigger or much smaller than the others. That unusual value is called an outlier.

Outliers are important because they can change how we understand a data set. In this lesson, you will learn how to spot an outlier and how it affects the mean and the median.

First, let’s review two measures of center.

• Mean: the average. Add all the data values, then divide by how many values there are.
• Median: the middle value when the data is put in order.

We often use the mean and median to describe the center of a data set. But they do not react the same way when there is an outlier.

What is an outlier?

An outlier is a data value that is far away from the other values in the set. It does not fit the general pattern of the data.

For example, in the data set \(4, 5, 5, 6, 6, 7, 30\), the number \(30\) is much larger than the other values. That makes \(30\) an outlier.

How can you identify an outlier?

In 6th Grade, a simple way to identify an outlier is to look for a value that is:

• much larger than most of the data, or
• much smaller than most of the data.

If nearly all the values are grouped together and one value is far away, that far-away value is likely an outlier.

Why do outliers matter?

Outliers can change the mean a lot because the mean uses every value in the data set.

The median usually changes very little, or not at all, because it only depends on the middle position of the ordered data.

This is why we say:

• The mean is affected strongly by outliers.
• The median is resistant to outliers.

Let’s prove this with numbers.

Suppose the data set is \(8, 9, 10, 11, 12\).

Find the mean:

$$ \text{Mean} = \frac{8+9+10+11+12}{5} = \frac{50}{5} = 10 $$

Find the median:

The data is already in order, and the middle value is \(10\).

So:

• Mean = \(10\)
• Median = \(10\)

Now add an outlier: \(50\).

The new data set is \(8, 9, 10, 11, 12, 50\).

New mean:

$$ \text{Mean} = \frac{8+9+10+11+12+50}{6} = \frac{100}{6} \approx 16.7 $$

New median:

There are now 6 numbers, so the median is the average of the 3rd and 4th values:

$$ \text{Median} = \frac{10+11}{2} = 10.5 $$

Look at what happened:

• The mean changed from \(10\) to about \(16.7\). That is a big jump.
• The median changed from \(10\) to \(10.5\). That is a small change.

This shows mathematically that the outlier pulled the mean much more than the median.

Worked Example 1: Spot the outlier

Data set: \(3, 4, 4, 5, 5, 6, 25\)

Step 1: Look for a value far from the others.

The numbers \(3, 4, 4, 5, 5, 6\) are close together. But \(25\) is far away.

So the outlier is \(25\).

Worked Example 2: Compare the mean and median with no outlier

Data set: \(6, 7, 8, 8, 9\)

Mean:

$$ \text{Mean} = \frac{6+7+8+8+9}{5} = \frac{38}{5} = 7.6 $$

Median:

The middle number is \(8\).

So:

• Mean = \(7.6\)
• Median = \(8\)

These two measures are close because the data values are close together.

Worked Example 3: Add an outlier and see the effect

Start with the same data: \(6, 7, 8, 8, 9\)

Add an outlier: \(30\)

New data set: \(6, 7, 8, 8, 9, 30\)

New mean:

$$ \text{Mean} = \frac{6+7+8+8+9+30}{6} = \frac{68}{6} \approx 11.3 $$

New median:

There are 6 numbers, so use the 3rd and 4th values:

$$ \text{Median} = \frac{8+8}{2} = 8 $$

Compare before and after:

• Mean changed from \(7.6\) to about \(11.3\).
• Median stayed at \(8\).

This is a strong example of how the mean is distorted by an outlier, while the median resists change.

Worked Example 4: A very small outlier

Data set: \(14, 15, 15, 16, 17\)

Original mean:

$$ \text{Mean} = \frac{14+15+15+16+17}{5} = \frac{77}{5} = 15.4 $$

Original median:

The middle value is \(15\).

Now add a very small outlier, \(1\).

New data set in order: \(1, 14, 15, 15, 16, 17\)

New mean:

$$ \text{Mean} = \frac{1+14+15+15+16+17}{6} = \frac{78}{6} = 13 $$

New median:

$$ \text{Median} = \frac{15+15}{2} = 15 $$

Compare:

• Mean changed from \(15.4\) to \(13\).
• Median stayed at \(15\).

So a small outlier can pull the mean downward, just like a large outlier can pull the mean upward.

How to decide whether the mean or median is better

If a data set has an outlier, the median is often a better way to describe the center because it is not strongly affected.

If the data does not have outliers and the values are fairly close together, the mean can work well.

Steps for solving problems about outliers

1. Put the data in order.
2. Look for any value far away from the others.
3. Find the mean and median.
4. Ask: Did one unusual value pull the mean up or down?
5. Compare how much the mean changed and how much the median changed.

Important idea to remember

The mean uses all the data values, so an extreme value changes the total sum a lot. Because of that, the mean can move a lot.

The median depends on the middle of the ordered list, so one extreme value at the end usually does not change the middle very much.

Quick Check

Consider the data set \(5, 6, 6, 7, 8, 40\).

• The outlier is \(40\).
• The median is the average of \(6\) and \(7\), which is \(6.5\).
• The mean is $$\frac{5+6+6+7+8+40}{6} = \frac{72}{6} = 12.$$

Notice that \(12\) is much larger than most of the values in the set. That tells us the mean is being pulled by the outlier.

Summary

An outlier is a value much larger or smaller than the rest of the data. Outliers can strongly change the mean because the mean uses every number in the set.

The median is usually more stable because it depends on the middle value or middle two values. So when a data set has an outlier, the median is often the better measure of center.