Skills

Descriptive Statistics

Descriptive statistics summarize a set of numbers.

Practice 3: Data Interpretation

Part 1 · Hook

Why this matters

A news story says the "average" reaction time in a driving study was 362 milliseconds. But on nine of ten trials the driver reacted in 320 ms or less; one trial was slow because the driver sneezed. Which number describes this driver? Choosing between the mean and the median is a question the exam asks again and again.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. Which kind of data is a list of reaction times in milliseconds?

  1. Quantitative data
  2. Qualitative data
  3. A Likert scale
  4. A case study
Show the answer

Numbers such as times, scores and counts are quantitative data.

  • Correct: Quantitative data:
  • Qualitative data:
  • A Likert scale:
  • A case study:

2. A scatterplot of hours studied against quiz scores shows points rising to the right. What does this show?

  1. A positive correlation
  2. A negative correlation
  3. No correlation
  4. That studying causes higher scores
Show the answer

Points rising together show a positive correlation, which does not by itself show cause.

  • Correct: A positive correlation:
  • A negative correlation:
  • No correlation:
  • That studying causes higher scores:

Part 4 · See it

See it first

A histogram of invented data: hours of sleep lost per week for 25 students. Most students lost 1 to 5 hours; a few lost 7, 8, 10 and 12, making a long tail to the right (positive skew). The mode is 3, the median is 4 and the mean is 4.32: the few high values pull the mean toward the tail, past the median.
In a positively skewed distribution, the mean is pulled toward the tail, past the median (invented data). LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. Add every value and divide by how many there arethe mean, which uses every value and so moves toward any outlier
  2. Put the values in order and take the middle one (or the average of the two middle ones)the median, which stays with the typical values even when there are outliers
  3. A few extreme values stretch one side of a distributionthe distribution is skewed and the mean moves toward the tail, so the median is the better summary
  4. Scores sit far from the meanthe range and the standard deviation are large: more variation

Part 6 · Key ideas

Key ideas

  • Central tendency: the mean (x¯\bar{x}, the total divided by the count), the median (the middle value in order) and the mode (the most frequent value). Two peaks make a bimodal distribution.
  • An outlier pulls the mean toward it. In a skewed distribution the mean sits toward the tail, so the median describes the typical case better.
  • Variation: the range is highest minus lowest; the standard deviation is roughly the typical distance of scores from the mean.
  • A frequency distribution counts how often each value or range occurs; a histogram shows it as touching bars.

Part 7 · Misconception

A common mistake

The wrong idea: The mean is always the best "average", and a negative skew means the scores are low.

What actually happens: With outliers or skew, the median is usually the better summary. Skew is named for the tail: a negative skew has a tail toward the low values, often because most scores are high.

Part 8 · Check yourself

Check yourself

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

1. Five students reported how many hours they slept the night before a test: 4, 7, 5, 8 and 6. What is the mean number of hours?

Type a number. No calculator needed: work it out by hand, as on the exam.

Show the answer

x¯\bar{x}=4+7+5+8+65= \frac{4 + 7 + 5 + 8 + 6}{5}=305= \frac{30}{5}=6= 6 hours.

  • Answer: 6

2. In a memory test, five participants recalled 3, 9, 4, 15 and 7 words. What is the median number of words recalled?

Type a number. No calculator needed: work it out by hand, as on the exam.

Show the answer

In order: 3, 4, 7, 9, 15. The middle (third) value is 7.

  • Answer: 7

3. A store reports the "average" income of its customers. A few customers are billionaires. Which measure would best describe a typical customer’s income?

  1. The mean, because it uses each customer’s income
  2. The median, because extreme values barely move it
  3. The range, because it covers the lowest and highest incomes
  4. The standard deviation, because it shows variation
Show the answer

Outliers pull the mean toward them. In a skewed distribution, the median better describes the typical case.

  • The mean, because it uses each customer’s income: Using every value is why the few billionaires pull it far up.
  • Correct: The median, because extreme values barely move it: Right: the median depends on the middle, not the extremes.
  • The range, because it covers the lowest and highest incomes: The range measures spread, not a typical value.
  • The standard deviation, because it shows variation: Standard deviation measures spread, not a typical value.

Data

Reaction times with an outlier

One participant’s reaction times on 10 trials, in milliseconds (invented data for practice). On one trial the participant sneezed.
Reaction time (ms)Number of trials
2802
3004
3203
9001

4. What is the mean reaction time across all 10 trials, in milliseconds?

Type a number (ms). No calculator needed: work it out by hand, as on the exam.

Show the answer

Total: 2(280)2(280)+4(300)+ 4(300)+3(320)+ 3(320)+900+ 900=560= 560+1200+ 1200+960+ 960+900+ 900=3620= 3620. Mean: 362010=362\frac{3620}{10} = 362 ms.

  • Answer: 362 ms

5. What is the median reaction time, in milliseconds?

Type a number (ms). No calculator needed: work it out by hand, as on the exam.

Show the answer

In order: 280, 280, 300, 300, 300, 300, 320, 320, 320, 900. The two middle values (5th and 6th) are both 300, so the median is 300 ms.

  • Answer: 300 ms

6. Which statistic gives the better summary of this participant’s usual reaction time, and why?

  1. The median, because the 900 ms sneeze trial pulls the mean up
  2. The mean, because it uses each trial, including the 900 ms one
  3. The range, because it shows the fastest and slowest trials
  4. The mode and the mean are equal, so either works
Show the answer

The median (300 ms) sits with the typical trials; the mean (362 ms) is higher than nine of the ten trials because of a single outlier.

  • Correct: The median, because the 900 ms sneeze trial pulls the mean up: Right: one outlier moves the mean far more than the median.
  • The mean, because it uses each trial, including the 900 ms one: Including the outlier is exactly why the mean misleads here.
  • The range, because it shows the fastest and slowest trials: The range describes spread, not a usual value.
  • The mode and the mean are equal, so either works: The mode is 300 and the mean is 362; they are not equal.

Part 9 · Summary

Summary

Descriptive statistics summarize a set of numbers. The mean, median and mode describe the center; the range and standard deviation describe the spread. Outliers pull the mean toward them, so in a skewed distribution the median is the better summary of a typical value. A frequency distribution or histogram shows the shape: symmetric, skewed toward a tail, or bimodal with two peaks.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

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