Descriptive statistics & distributions
The idea
Descriptive statistics compresses a dataset into a few honest numbers and pictures: measures of center (mean, median), measures of spread (range, interquartile range, standard deviation), and the distribution's shape — symmetric, skewed, or studded with outliers. You have computed means and read plots before; the upgrade here is the standard deviation, the typical distance of data points from their mean, and the z-score, which counts how many standard deviations a value sits from the mean and puts scores from different scales on one comparable footing.
Match the summary to the shape: the mean and standard deviation suit roughly symmetric data, but both chase outliers, so skewed data is served better by the median and IQR. To compute a sample standard deviation: find the mean, square each deviation from it, average the squares dividing by n − 1, and take the square root. The misconception to correct is that center tells the whole story — two classes can share a mean of 82 while one ranges from 80 to 84 and the other from 60 to 100, which are utterly different teaching situations. Spread is information, not noise.
Worked example
Five quiz scores are 72, 78, 81, 85, and 94. Find the mean, the sample standard deviation, and the z-score of the 94, then interpret the z-score.
- Mean: (72 + 78 + 81 + 85 + 94)/5 = 410/5 = 82.
- Deviations from the mean: −10, −4, −1, 3, and 12. They sum to zero, a built-in check that the mean is correct.
- Square and total the deviations: 100 + 16 + 1 + 9 + 144 = 270. Divide by n − 1 = 4 to get the sample variance: 270/4 = 67.5.
- Take the square root for the standard deviation: √67.5 ≈ 8.22, so scores typically sit about 8 points from the mean.
- z-score of the top score: (94 − 82)/8.22 ≈ 1.46, meaning the 94 lies about one and a half standard deviations above the mean — strong, but not an extreme outlier.
Answer. Mean 82, sample standard deviation about 8.22, and the 94 has z ≈ 1.46 — roughly 1.5 standard deviations above average.
Check your understanding
- Why do deviations from the mean always sum to zero, and how does that fact force the squaring step in the variance?
- When would the median and IQR describe a dataset more honestly than the mean and standard deviation?
- What comparisons does a z-score make possible that raw scores cannot, and what assumptions hide inside such comparisons?
- How would adding a single score of 40 to this dataset change the mean, the median, and the standard deviation differently?
Build the foundations first
Descriptive statistics & distributions builds on these concepts. If any feel shaky, start there.