Descriptive Statistics

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Skill: descriptive-statistics

Descriptive Statistics

Summarize data with measures of center and spread, pick the right graph, and describe distributions with SOCS.

1 Center: mean, median, mode

The center is a single number that represents the “typical” value of a dataset. The three common measures behave differently when data is skewed.

Definitions
mean x̄ = (sum of values) / n   |   median = middle value of sorted data

Mean uses every value, so outliers and skew pull it. Median is the middle of the sorted list, so it resists outliers. Mode is the most frequent value (best for categories).

Example Quiz scores: 70, 74, 78, 80, 98
  1. Mean: (70 + 74 + 78 + 80 + 98) / 5 = 400 / 5 = 80.
  2. Median: the sorted middle value is 78.
  3. The high score 98 pulls the mean above the median — a hint of right skew.
Example House prices with one mansion
  1. Prices ($k): 320, 340, 350, 360, 2900. Mean = 4270 / 5 = 854.
  2. Median = 350 — far more typical of the neighborhood.
  3. Report median for heavily skewed data like incomes and house prices.

2 Spread: range, IQR, standard deviation

Spread says how far values stray from the center. Range is quick but fragile; IQR and standard deviation are the workhorses.

Definitions
range = max − min   |   interquartile range (IQR) = Q3 − Q1   |   s = √( Σ(x − x̄)² / (n − 1) )

Quartiles split sorted data into quarters: Q1 is the median of the lower half, Q3 the median of the upper half. The IQR covers the middle 50% and resists outliers. Standard deviation is the root-mean-square distance from the mean — not the average distance.

min 4 Q1 5 median 8 Q3 11 max 12 IQR = 11 − 5 = 6 (the box)
Example Data: 4, 6, 8, 10, 12
  1. Range = 12 − 4 = 8. Median = 8; lower half {4, 6} → Q1 = 5; upper half {10, 12} → Q3 = 11.
  2. IQR = 11 − 5 = 6.
  3. Mean = 8; squared deviations: 16, 4, 0, 4, 16; sum = 40; sample SD = √(40/4) = √10 ≈ 3.16.

3 Shape, skew, and choosing a graph

The shape of a distribution tells you which center/spread pair to report, and the variable type tells you which graph to draw.

Skew rule of thumb
right skew (tail right): mean > median   |   left skew (tail left): mean < median

The mean chases the tail. For skewed data report median and IQR; for roughly symmetric data report mean and standard deviation.

Example Mean 62, median 71
  1. Mean < median, so the tail stretches left: skewed left.
  2. Report the median (71) and IQR here, not the mean.
Example Picking a graph
  1. Favorite fruit (categorical) → bar chart.
  2. Heights of 200 students (quantitative) → histogram.
  3. Comparing test scores across three classes → side-by-side boxplots.

4 SOCS: describe any distribution

SOCS is the four-point checklist statisticians use: Shape, Outliers, Center, Spread — always in context.

SOCS checklist
Shape (symmetric? skewed? modes?) · Outliers (any? how far?) · Center (mean or median) · Spread (SD or IQR)

Shape first: it decides whether center/spread should be mean/SD or median/IQR.

Example “Ages at a playground”
  1. Shape: right-skewed (many kids, a few adults). Outliers: a couple of grandparents near 70.
  2. Center: median age ≈ 8. Spread: IQR ≈ 6 years.
  3. Full sentence: “Ages are right-skewed with a few high outliers; the typical age is about 8 (median), with the middle 50% spanning about 6 years.”

5 Common mistakes

Mean-only reporting for skewed data. Incomes, house prices, and waiting times are usually right-skewed — the mean overstates the typical value. Pair it with the median, or report median/IQR instead.
“Average deviation” confusion. Standard deviation is the root-mean-square distance from the mean, not the average of |x − x̄|. The squaring penalizes far values more.
Histogram for categorical data. Bars must touch only for quantitative data. Categories get separated bars (bar chart); a pie chart works too for parts of a whole.

6 Key vocabulary

Mean (x̄)
Sum divided by n; pulled by skew and outliers.
Median
Middle of sorted data; resistant to outliers.
Quartiles / IQR
Q1, Q3 split the data in quarters; IQR = Q3 − Q1 covers the middle 50%.
Standard deviation
Root-mean-square distance from the mean; use n − 1 for samples.
Skew
Direction of the long tail; the mean chases the tail.
SOCS
Shape, Outliers, Center, Spread — in context.

7 Quick checks

1. Mean 90, median 74 — which skew?
Right-skewed. Mean > median means the tail stretches right.
2. Which pair resists outliers: mean/SD or median/IQR?
Median and IQR. Both are built from order, not sums.
3. Data: 4, 6, 8, 10, 12 — what are Q1, Q3, and the IQR?
Q1 = 5, Q3 = 11, IQR = 6. Split at the median 8; each half’s median gives the quartile.
Practice Stats calculator Mean/median lab
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