Describe distributions using SOCS: Shape (symmetric, skewed left/right, bimodal, uniform), Outliers (presence and impact), Center (mean or median), Spread (standard deviation or IQR). Always write in context.
A lesson of AP Statistics, in Exploring One-Variable Data, about 10 minutes.
Exploring Data: Describing Distributions
By the end of this lesson, you will be able to describe and compare distributions using appropriate graphical displays and summary statistics, identify shape, center, spread, and outliers, and interpret distributions in context.
The Language of Data
Variable types:
Categorical (qualitative): records which category an individual belongs to (color, gender, yes/no)
Quantitative: records numerical measurements (height, SAT score, temperature)
Discrete: countable values (number of pets)
Continuous: any value in an interval (weight, time)
Graphical Displays
For Categorical Data
Bar chart: bars represent counts or percents; bars are separated (not touching). Order can vary (but often sorted by frequency).
Pie chart: shows relative proportions of a whole; all categories together = 100%.
For Quantitative Data
Dotplot: each data value is a dot; good for small datasets; easy to see shape.
Stemplot (Stem-and-Leaf): stem = leading digit(s); leaf = last digit. Shows actual values AND shape.
Histogram: groups data into equal-width intervals ("bins"); height = frequency or relative frequency; bars touch (continuous data). Good for large datasets.
Boxplot (Box-and-Whisker): displays the five-number summary.
Box spans Q₁ to Q₃ (IQR)
Line at median
Whiskers extend to the smallest/largest non-outlier
Outliers plotted individually
Describing Distributions: SOCS
When describing a distribution, address all four:
Shape
Outliers
Center
Spread
Always interpret in context (mention the variable and population).
Shape
Symmetric: roughly equal on both sides; mean ≈ median
Right-skewed (positively skewed): tail extends to the right; mean > median (outliers pull the mean)
Left-skewed (negatively skewed): tail extends to the left; mean < median
Unimodal: one peak; bimodal: two peaks; uniform: roughly flat
Outliers
IQR method (1.5 × IQR rule):
$\text{Outlier if}: x < Q_1 - 1.5 \times IQR \quad \text{or} \quad x > Q_3 + 1.5 \times IQR$
Measures of Center
Mean ($\bar{x}$): $\bar{x} = \dfrac{\sum x_i}{n}$; uses all values; sensitive to outliers.
Median ($M$): middle value of ordered data; resistant to outliers.
Choosing between mean and median:
Symmetric, no outliers: either; both equal approximately
Skewed or outliers: median is a better representative measure
Measures of Spread
Range: $\text{Max} - \text{Min}$; simple but affected by outliers.
Interquartile Range (IQR): $Q_3 - Q_1$; the range of the middle 50%; resistant to outliers. Used with the median.
Standard Deviation ($s$):
$$s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}}$$
Measures the typical distance of values from the mean. Used with the mean.
Variance ($s^2$): the square of the standard deviation.
Properties of standard deviation:
Always ≥ 0; equals 0 only if all values are identical
Sensitive to outliers
Has the same units as the data (unlike variance)
Choosing between IQR and standard deviation:
Symmetric, no outliers: use mean and standard deviation
Skewed or outliers: use median and IQR
The Five-Number Summary
Five-Number Summary: Min, Q₁, Median, Q₃, Max
Used to construct a boxplot. Gives a quick summary of the distribution's shape and spread.
Example: Data: 2, 5, 7, 8, 9, 12, 15, 18, 20, 25
$n = 10$; ordered already.
Median = average of 5th and 6th values: $(9+12)/2 = 10.5$