Math Class Didn’t Show Its Work, companion reader · Chapter 24

The Average Is Hiding Something

An average is a useful witness. It is not the whole story.

These four demonstrations put the chapter's small lists on the table one observation at a time. Keep every repeat, sort before you look for a middle, and watch which summaries move when one value changes and which stay put.

Every example in these readers is a constructed teaching example taken from or modeled on the book's own worked examples. The numbers are declared inputs chosen to make the mathematics visible. They are not measurements of any real person, product or study.

Demonstration 1 of 4

Every repeat is a person

Three waiting times are 3, 3 and 8. What happens to the mean if a repeat is dropped?

Each dot is one observation. Adding a copy of 3 adds 3 to the sum and 1 to the count, so the balance point slides toward 3.

\[(3+3+8)/3=14/3\]

\[(3+8)/2=5.5\]

Square brackets hold a data list, where repeats count. The mean is the sum of the listed values divided by how many there are.

Predict first. Before choosing one copy of 3, guess whether the mean goes up or down.

Choose an example

Figure: Every repeat is a person. A dot plot of waiting times with 2 stacked dots at 3 and one dot at 8. A triangle marks the mean at 14/3, about 4.67.
Copies of the waiting time 3: 2
Constructed example: the chapter's list [3, 3, 8], with other numbers of repeated 3s.

Calculated values

List
[3, 3, 8]
Number of observations
3
Sum
3 + 3 + 8 = 14
Mean
14/3, about 4.67

Mean = (3 + 3 + 8)/3 = 14/3, about 4.67. This is the book's list. Dropping one 3 would give (3 + 8)/2 = 5.5 instead.

Use the idea

When you average a list, count the observations you are dividing by. Two equal waits are still two people waiting.

Where the conclusion applies

A constructed list of waiting times, with no unit stated, where every observation counts once. Treating the list as a set would silently drop repeats and answer a different question.

Check your understanding: What is the mean of [5, 5, 5, 9]?
(5 + 5 + 5 + 9)/4 = 24/4 = 6.

Chapter 24 source: section "Repeated observations stay in the list". Demonstration C24-D01.

Demonstration 2 of 4

Same mean, different data

Datasets A and B both have mean 6. Are they the same data?

Both datasets total 30 across 5 values, so sharing out gives 6 everywhere. The mean is that leveled picture. The raw bars keep what the mean leaves out: how the values are arranged.

\[\text{mean}=\dfrac{\text{sum of values}}{\text{number of values}}\]

\[\text{mean}(B) = (0 + 0 + 6 + 12 + 12)/5\]

A = [2, 4, 6, 8, 10] and B = [0, 0, 6, 12, 12]. Each bar is one value in its listed position. The leveled view shares the total out equally across the five positions.

Predict first. Switch to the leveled view for A, then for B. Will the two pictures differ?

Choose an example

Figure: Same mean, different data. Five bars for Dataset A, with heights 2, 4, 6, 8, 10. A dashed line marks the mean, 6.
Dataset: A = [2, 4, 6, 8, 10], View: Values as listed
Constructed example: the chapter's Datasets A and B and Figure 24.1.

Calculated values

Dataset A
[2, 4, 6, 8, 10]
Sum
2 + 4 + 6 + 8 + 10 = 30
Count
5
Mean
30/5 = 6
Values 6 units from the mean
0

mean(A) = (2 + 4 + 6 + 8 + 10)/5 = 30/5 = 6. The values step away from 6 in both directions.

Use the idea

When a report gives only an average, ask for the count, the unit and the raw values or a faithful picture of them.

Where the conclusion applies

Two constructed lists of five values each. The mean is exact; calling 6 a typical value is an extra interpretation that fits A better than B.

Check your understanding: What is the mean of [1, 5, 6, 7, 11]?
(1 + 5 + 6 + 7 + 11)/5 = 30/5 = 6, the same center again with yet another arrangement.

Chapter 24 source: section "A dataset comes before its summary". Demonstration C24-D02.

Demonstration 3 of 4

Sort first, then find the middle

Waiting times were recorded as 9, 2, 7, 3, 4. What is the median?

Sorting moves the cards but keeps every value, so the sum and count do not change. Only after sorting does the middle position mean something.

\[(9+2+7+3+4)/5=25/5=5\]

\[(4+6)/2=5\]

The median is the middle value after sorting. With an even count it is the mean of the two middle values. The range is largest minus smallest.

Predict first. Look at the recorded order for the five times. Does its middle card give the median?

Choose an example

Figure: Sort first, then find the middle. 5 cards in recording order: 9, 2, 7, 3, 4. The middle position is outlined, which is not the median until the list is sorted.
List: [9, 2, 7, 3, 4], Order: As recorded
Constructed example: the chapter's lists [9, 2, 7, 3, 4] and [8, 2, 6, 4].

Calculated values

Recorded
[9, 2, 7, 3, 4]
Sorted
[2, 3, 4, 7, 9]
Median
4
Mean
25/5 = 5
Range
9 - 2 = 7

Picking the middle of the unsorted record gives 7, which is not the median. After sorting to [2, 3, 4, 7, 9], the middle (third) value is 4. Mean = (9 + 2 + 7 + 3 + 4)/5 = 25/5 = 5. Sorting keeps the sum 25 and the count 5.

Use the idea

Before reporting a median, sort a copy of the list and check that its total matches the original.

Where the conclusion applies

Small constructed lists: the five waiting times are in minutes, and the four-value list has no stated unit. With an even count the median need not be one of the observations.

Check your understanding: What is the median of [10, 1, 7, 4, 3, 12]?
Sorted: [1, 3, 4, 7, 10, 12]. The central pair is 4 and 7, so the median is (4 + 7)/2 = 5.5.

Chapter 24 source: section "Worked example: sort first, then choose the middle". Demonstration C24-D03.

Demonstration 4 of 4

One high value pulls the mean

In [1, 1, 1, 1, 21], why are the mean and the median so far apart?

The four 1s fix the median at 1 no matter how large the last value is. The mean and the range both grow with the high value, because they notice its size.

\[(1 + 1 + 1 + 1 + 21)/5 = 25/5 = 5\]

\[21 - 1 = 20\]

The mean uses every value through the total. The median is the third value in order. The range is the largest value minus the smallest.

Predict first. Change the high value from 21 to 101. Which of mean, median and range stay the same?

Choose an example

Figure: One high value pulls the mean. A dot plot of [1, 1, 1, 1, 21]. The median mark sits at 1 and the mean triangle at 5.
The fifth value: 21
Constructed example: the chapter's list [1, 1, 1, 1, 21] and its 101 variation.

Calculated values

List
[1, 1, 1, 1, 21]
Mean
25/5 = 5
Median
1
Range
21 - 1 = 20

Mean = (1 + 1 + 1 + 1 + 21)/5 = 25/5 = 5. Median = third ordered value = 1. Range = 21 - 1 = 20. The single high value pulls the mean from 1 up to 5; the median stays at 1 because four of the five ordered positions hold 1.

Use the idea

When mean and median disagree, report both and look at the list before deciding what the gap means.

Where the conclusion applies

A constructed list. The chapter calls 21 a single high value, not an outlier, because no outlier rule has been stated. Deleting it would change the data, not clean them.

Check your understanding: For [1, 1, 1, 1, 101], what are the mean, median and range?
Mean (1 + 1 + 1 + 1 + 101)/5 = 105/5 = 21, median 1, range 101 - 1 = 100.

Chapter 24 source: section "One high value can move the mean". Demonstration C24-D04.