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D1.5 · Explore how adding or removing data changes averages and distributions

Learn to explore how adding or removing data changes averages and distributions through clear examples and targeted practice.

Ontario Grade 7 Mathematics

Data

Explore what happens when values are added to or removed from a data set

Imagine four students record how many minutes they read one evening: 4, 5, 5, and 6. Then a fifth student records 12 minutes. What happens to the averages? What happens to the way the values are arranged? A data set is a group of recorded values. An average is a number that helps describe a data set. The mean and median are two averages. A distribution describes the values and how they are arranged, including how often they appear and how spread out they are. We can compare the data before and after a change to explore all three ideas.

What you will learn

Find and compare the mean and median before and after data changes.
Describe how adding or removing values changes a data set's distribution.
Use calculations and a chart to explain what changed and what stayed the same.

Review the mean and median

To find a mean, add the values to get a total. Count how many values there are. Divide the total by that count. The mean can be thought of as a fair share: if the total were shared equally, each value would equal the mean.

For the reading times 4, 5, 5, and 6, the total is 20 and there are 4 values. The mean is 5 minutes. This calculation uses every value.

To find a median, first put the values in order from least to greatest. The median is the middle value. When there are two middle values, the median is halfway between them. In the reading-time list, the two middle values are both 5, so the median is 5 minutes.

After adding or removing a value, find the mean again using the new total and new count. Put the changed list in order and find its median again. The number of values may change, so the middle position may also change.

A value far from most of the others can affect the mean because every value is used in the total. The median depends on position in the ordered list. The two averages can therefore respond differently to the same change.

Mean: add the values and divide by how many values there are.
Median: order the values and find the middle, or the halfway point between the two middle values.
After a change, recount and check the new ordered list.

Notice changes in a distribution

A distribution describes the values in a data set and how they are arranged. You can notice which values appear often, which appear rarely, and whether the values are close together or spread out. These observations help describe more than an average alone.

A bar chart can show how often each value occurs. The number of times a value occurs is its frequency. For example, a bar of height 2 above the value 5 means that 5 occurs twice. The position along the bottom shows the value; the height shows its frequency.

For the reading times 4, 5, 5, and 6, the frequencies are one 4, two 5s, and one 6. Adding 12 gives the data a new value beyond the group from 4 to 6. On a chart, a new bar appears at 12. The chart makes that change easy to see.

Sometimes a distribution changes even though the mean stays the same. For example, adding a value already common in the data can increase its frequency. The mean alone does not show every detail of a distribution.

A distribution describes which values occur and how they are arranged.
Frequency means how many times a value occurs.
A chart can show a change that an average does not reveal.

Predict, calculate, and compare

Before calculating, you can predict how adding one value will affect the mean. Compare the added value with the old mean. A value above the old mean pulls the mean upward. A value below the old mean pulls it downward. A value equal to the old mean leaves the mean unchanged.

For example, the mean of 3, 5, and 7 is 5. Adding another 5 leaves the mean at 5. The total and the count both increase, but the mean does not. The distribution still changes because 5 occurs more often.

Removing a value has the reverse effect on the mean. Removing a value above the old mean lowers the mean of the remaining values. Removing one below the old mean raises it. In either case, recalculate to find the new mean. Do not rely on a prediction alone.

The median may change or stay the same when a value is added or removed. Check the ordered list each time. A useful comparison reports the old and new averages, then describes the change in the values or their frequencies.

Compare an added value with the old mean to predict the direction of change.
For a removed value, the direction of the mean change is reversed.
Recalculate the mean and find the median from the changed list.

A reliable way to explain a change

Use this simple order: write the original data, make only the stated change, and write the new data. Find the old and new means using the total and count for each list. Order each list to find its median. Then compare the values or use a chart to describe the distribution.

Use evidence in your explanation. You might say, “The mean increased because the added value was above the original mean.” For the distribution, name what changed: a new value appeared, a value occurred more often, or the values now reach farther toward one end.

Do not assume that one average tells the whole story. Two lists may have the same mean but arrange their values differently. A clear explanation separates the numerical averages from the description of the distribution.

Change only the data specified in the question.
Use the new total and count for the new mean.
Support your description of the distribution with the actual values or a chart.

Worked example

Adding a value above the mean

Hypothetical reading times, in minutes, are 4, 5, 5, and 6. A fifth student records 12 minutes. Compare the mean, median, and distribution before and after the addition.

Reading times after the addition

Hypothetical reading-time data from the example. Each bar height gives the frequency of that time.

Find the original averages
Add the four original times to get 20. Divide by 4 to find a mean of 5 minutes. The middle two values in order are both 5, so the median is 5 minutes.
$20\div4=5$
Add 12 minutes
The new total is 32 and the new count is 5. Divide to find a new mean of 6.4 minutes. The ordered list is 4, 5, 5, 6, 12, so its middle value is 5 minutes.
$32\div5=6.4$
Describe the change
The mean rises from 5 to 6.4 minutes because the added value, 12, is above the old mean. The median stays at 5 minutes. The distribution now includes a value beyond the group from 4 to 6.

Answer: The mean changes from 5 minutes to 6.4 minutes. The median stays at 5 minutes. The added value extends the distribution to 12 minutes.

Check: The new mean, 6.4 minutes, is between the smallest value, 4, and the largest value, 12.

Worked example

Removing the largest value

Hypothetical page counts are 3, 4, 4, 5, and 14. The value 14 is removed because it was entered by mistake. Find how the mean, median, and distribution change.

Page counts before removal

Hypothetical page-count data before 14 is removed. Each bar height gives the frequency of that count.

Find the original averages
The five values total 30, so the mean is 6 pages. The middle value in the ordered list is 4, so the median is 4 pages.
$30\div5=6$
Remove 14
The remaining values total 16, and there are 4 of them. Their mean is 4 pages. The two middle values are both 4, so the new median is also 4 pages.
$16\div4=4$
Compare the distribution
The mean falls from 6 to 4 pages, while the median stays at 4 pages. The remaining values range from 3 to 5. There is no longer a value at 14.

Answer: Removing 14 lowers the mean from 6 pages to 4 pages. The median remains 4 pages, and the remaining values are grouped from 3 to 5.

Check: The remaining values add to 16. Dividing by the 4 remaining values gives a mean of 4 pages.

Worked example

Changing frequency without changing the mean

A data set is 3, 5, and 7. Two values, 5 and 5, are added. Compare the mean and median before and after the addition. Describe the distribution.

Values after adding two 5s

Hypothetical data after the addition. Each bar height gives the frequency of that value.

Find the original averages
The original total is 15 across 3 values, so the mean is 5. The middle value is also 5, so the median is 5.
$15\div3=5$
Add two 5s
The new total is 25 across 5 values, so the mean remains 5. The ordered list is 3, 5, 5, 5, 7. Its middle value is 5, so the median remains 5 too.
$25\div5=5$
Explain the distribution
Both averages stay the same, but the distribution changes. The value 5 occurred once before and occurs three times now. More of the data are grouped at 5.

Answer: The mean and median both remain 5. The distribution changes because 5 occurs more often.

Check: Each added value equals the original mean, 5. The new total and count still give a mean of 5.

Common mistakes and how to avoid them

Dividing the new total by the old number of values.

Correction: Recount after adding or removing data. Use the new count to find the new mean.

Assuming the median changes whenever the mean changes.

Correction: Order the changed data and find the middle again. The median may stay the same.

Saying the distribution is unchanged because the mean is unchanged.

Correction: Check which values occur and how often. The mean does not show every detail of the distribution.

Lesson summary

The mean is the total divided by the number of values.
The median is the middle value in an ordered list, or halfway between the two middle values.
Adding or removing data can change the mean, the median, the distribution, or more than one of these.
A chart and a calculation together give a clearer comparison than an average alone.

Check your understanding

Question 1

The data are 2, 4, and 6. A value of 12 is added. What happens to the mean?

It rises from 4 to 6.
It stays at 4.
It falls from 4 to 3.
correctIndex Is Not Used Here?

Show answer and explanation

It rises from 4 to 6.

The original total is 12 across 3 values, so the mean is 4. The new total is 24 across 4 values, so the new mean is 6.

Question 2

The data are 4, 5, and 6. Another 5 is added. What happens to the mean?

It rises above 5.
It stays at 5.
It falls below 5.
correctIndex Is Not Used Here?

Show answer and explanation

It stays at 5.

The original mean is 5. The added value equals that mean, so the new total and count still give a mean of 5.

Question 3

A data set has values 2, 3, 3, and 10. The 10 is removed. Which statement is correct?

The mean decreases, and the remaining values are closer together.
The mean increases, and the remaining values are farther apart.
The mean and distribution must stay exactly the same.
correctIndex Is Not Used Here?

Show answer and explanation

The mean decreases, and the remaining values are closer together.

The original mean is 4.5. After removing 10, the remaining total is 8 across 3 values, so the mean is about 2.67. The remaining values range only from 2 to 3.

Key terms

Data set: A group of recorded values.
Mean: The total of the values divided by the number of values.
Median: The middle value in an ordered data set, or halfway between the two middle values.
Distribution: The values in a data set and how they are arranged, including how often they occur and how spread out they are.
Frequency: How many times a value occurs.

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