Unit 1 · Topic 1.7 Beta

Summary Statistics for One Quantitative Variable

The mean and standard deviation use every value, so outliers and skew pull them; the median and IQR are resistant.

Practice 3: Analyze DataPractice 4: Interpret Results

Part 1 · Hook

Why this matters

A reviewer tests 12 phones of one model. Eleven last between 7 and 11 hours; one lasts 14.9. Report the mean and the typical phone seems to last 9.68 hours. Report the median and it is 9.4. Both are correct. Which one to report depends on what each summary does with that one long-lasting phone, and that is the heart of this topic.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. Evaluate $\frac{3 + 5 + 10}{3}$ .

6
11.33
18
3

Show the answer

Add the top first: 18, then divide by 3.

Correct: 6:
11.33:
18:
3:

2. Which shape has a long tail toward the larger values?

Skewed right
Skewed left
Uniform
Symmetric

Show the answer

Skew is named for the tail.

Correct: Skewed right:
Skewed left:
Uniform:
Symmetric:

3. What is $\sqrt{\frac{26}{2}}$ , to two decimal places?

3.61
2.55
13
6.5

Show the answer

$\frac{26}{2} = 13$ and $\sqrt{13} = 3.61$ .

Correct: 3.61:
2.55:
13:
6.5:

Part 4 · See it

See it first

Adding one phone that lasts 14.9 hours moves the mean from 9.21 to 9.68 hours, but the median only from 9.30 to 9.40. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

The mean adds every value and divides by none extreme value pulls it toward the tail
The median and quartiles depend only on the order of the valuesthey barely move when an extreme value changes: they are resistant
The standard deviation squares each distance from the meanit measures typical distance from the mean, and an outlier inflates it
So for skewed data or outliersreport the median and IQR; for roughly symmetric data, the mean and standard deviation

Part 6 · Key ideas

Key ideas

Mean $\bar{x} = \frac{\sum x_i}{n}$ ; median: the middle of the ordered values (the average of the two middle ones when n is even).
Quartiles: $Q_1$ and $Q_3$ are the medians of the lower and upper halves (leave out the median when n is odd). $\text{IQR} = Q_3 - Q_1$ ; range = max − min.
Standard deviation $s$ $= \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}$ , the typical distance from the mean; variance is $s^2$ . Population versions: $\mu$ and $\sigma$ .
Outliers: beyond $Q_1 - 1.5 \times \text{IQR}$ or $Q_3 + 1.5 \times \text{IQR}$ , or more than 2 standard deviations from the mean.
Adding a constant shifts the center but not the spread; multiplying by a constant multiplies both.

Part 7 · Misconception

A common mistake

The wrong idea: The mean is always the best measure of center, because it uses every value.

What actually happens: Using every value is what makes the mean sensitive to extreme values. With strong skew or outliers, the resistant median describes a typical value better.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Graph

Battery life of 12 phones

A tech reviewer ran 12 phones of one model on the same video until the battery died. The battery lives, in hours, were 7.2, 8.1, 8.4, 8.9, 9.0, 9.3, 9.5, 9.8, 10.1, 10.4, 10.6 and 14.9 (invented data).

1. What is the mean battery life, in hours?

Type a number, in hours. A table answer and a calculator answer are both accepted.

Show the answer

$\bar{x}$ $= \frac{\sum x_i}{n}$ $= \frac{116.2}{12}$ $= 9.68$ hours.

Answer: 9.68 hours

2. What is the median battery life, in hours?

Type a number, in hours. A table answer and a calculator answer are both accepted.

Show the answer

With $n = 12$ , the median is the average of the 6th and 7th ordered values: $\frac{9.3 + 9.5}{2} = 9.4$ hours.

Answer: 9.4 hours

3. What is the interquartile range (IQR) of the battery lives, in hours?

Type a number, in hours. A table answer and a calculator answer are both accepted.

Show the answer

Lower half 7.2 to 9.3: $Q_1$ $= \frac{8.4 + 8.9}{2}$ $= 8.65$ . Upper half 9.5 to 14.9: $Q_3$ $= \frac{10.1 + 10.4}{2}$ $= 10.25$ . $\text{IQR} = 10.25 - 8.65 = 1.6$ hours.

Answer: 1.6 hours

4. What is the standard deviation of the battery lives, in hours? Use the sample standard deviation.

Type a number, in hours. A table answer and a calculator answer are both accepted.

Show the answer

$s$ $= \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}$ $= \sqrt{\frac{40.34}{11}}$ $= 1.91$ hours. Technology gives the same value.

Answer: 1.91 hours

5. Which battery lives are outliers by the $1.5 \times \text{IQR}$ rule?

7.2 and 14.9 hours
14.9 hours alone
No phone, since 14.9 is below twice the mean of 9.68 hours
10.6 and 14.9 hours

Show the answer

Fences: $Q_1 - 1.5 \times \text{IQR} = 6.25$ and $Q_3 + 1.5 \times \text{IQR} = 12.65$ . Only 14.9 lies outside.

7.2 and 14.9 hours: 7.2 is above the lower fence of 6.25, so it is not an outlier.
Correct: 14.9 hours alone: Right: the fences are $8.65 - 1.5(1.6) = 6.25$ and $10.25 + 1.5(1.6) = 12.65$ ; only 14.9 is outside.
No phone, since 14.9 is below twice the mean of 9.68 hours: Twice the mean is not an outlier rule. Use the fences: 14.9 is above 12.65.
10.6 and 14.9 hours: Values above $Q_3$ are common; an outlier must be beyond the fence, 12.65.

6. A student’s score is at the 80th percentile of a test. What does that tell you?

80% of the scores are at or below this score
The student answered 80% of the questions correctly
The score was 80 points
80 students scored higher than this student on the test

Show the answer

The pth percentile is the value with p% of the data at or below it.

Correct: 80% of the scores are at or below this score: Right: a percentile tells the percent of the data at or below a value.
The student answered 80% of the questions correctly: A percentile is about position among scores, not the percent correct.
The score was 80 points: A percentile is not a score; it tells where a score stands.
80 students scored higher than this student on the test: A percentile is a percent of the data below, not a count above.

7. Which summaries are resistant to an extreme outlier? Select all that apply.

The median
The IQR
The mean
The standard deviation
The range

Show the answer

The median and IQR are resistant; the mean, standard deviation and range are not.

Correct: The median: Right: it depends on the middle of the ordered data.
Correct: The IQR: Right: it depends on the quartiles, not the extremes.
The mean: The mean adds in the outlier, so it moves toward it.
The standard deviation: An outlier far from the mean makes one squared distance huge.
The range: The range uses the maximum or minimum, which may be the outlier.

Part 9 · Summary

Summary

The mean and standard deviation use every value, so outliers and skew pull them; the median and IQR are resistant. Quartiles are the medians of the two halves, and the

1.5 \times \text{IQR}

fences or the 2 standard deviation rule flag outliers. Adding a constant shifts center but not spread; multiplying by a constant scales both.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

Question set for this topic