Data Displays & Distribution Shape

Why visualize?

Summary numbers hide structure. Two data sets can share the exact same mean and look completely different. A picture reveals shape, center, spread, and oddities at a glance — before you trust any single number.

Histograms

A histogram groups values into intervals (bins) and draws a bar for how many values land in each bin. A tall bar means many values in that range. The outline shows the shape of the distribution.

Common shapes:

• Symmetric (often bell-shaped) — balanced around the center.
• Right-skewed — a long tail stretching to the right.
• Left-skewed — a long tail stretching to the left.
• Bimodal — two distinct peaks (often two groups mixed together).

Skew drags the mean

The skew direction is the direction of the long tail. A few extreme values in the tail pull the mean toward them, while the median barely moves:

• Right-skewed: $\text{mean} > \text{median}$
• Left-skewed: $\text{mean} < \text{median}$
• Symmetric: $\text{mean} \approx \text{median}$

Income is the classic right-skewed example: a few huge earners pull the mean above the median.

Quartiles and the five-number summary

Sort the data. Quartiles cut it into four equal parts:

• $Q_1$ (25th percentile): the median of the lower half.
• $Q_2$ (50th percentile): the median itself.
• $Q_3$ (75th percentile): the median of the upper half.

The five-number summary is: minimum, $Q_1$, median, $Q_3$, maximum.

Example. Data: $2, 4, 5, 7, 8, 10, 12, 14, 18$ (n = 9). The median is the 5th value, $8$. Lower half $\{2,4,5,7\}$ gives $Q_1 = \frac{4+5}{2} = 4.5$. Upper half $\{10,12,14,18\}$ gives $Q_3 = \frac{12+14}{2} = 13$.

Five-number summary: $2,\ 4.5,\ 8,\ 13,\ 18$.

Interquartile range (IQR)

$\text{IQR} = Q_3 - Q_1$

The IQR is the spread of the middle 50% of the data. Because it ignores the extremes, it's resistant to outliers. For the example: $\text{IQR} = 13 - 4.5 = 8.5$.

Box plots

A box plot draws the five-number summary: a box from $Q_1$ to $Q_3$ with a line at the median, and whiskers extending to the smallest and largest non-outlier values. Great for comparing several groups side by side.

The 1.5 × IQR rule for outliers

A value is flagged as an outlier if it falls below the lower fence or above the upper fence:

$\text{lower fence} = Q_1 - 1.5 \times \text{IQR}$ $\text{upper fence} = Q_3 + 1.5 \times \text{IQR}$

Example. Lower fence $= 4.5 - 1.5(8.5) = -8.25$; upper fence $= 13 + 1.5(8.5) = 25.75$. Every value sits inside, so there are no outliers.

Choosing center and spread by shape

• Symmetric, no outliers → report the mean and standard deviation.
• Skewed or outliers present → report the median and IQR (the resistant measures).

Key takeaways

• Histograms reveal shape — watch for skew and multiple peaks.
• Skew points toward the long tail and drags the mean, not the median.
• Five-number summary builds the box plot; $\text{IQR} = Q_3 - Q_1$.
• Flag outliers with the $1.5 \times \text{IQR}$ rule.
• Skew or outliers? Prefer the median and IQR.