2. Exploring Data

b. Numerical Data

Yurk

Michigan County Life Expectancy

Our running example for numerical data:

  • 83 Michigan counties
  • Variable: life expectancy (years)

Numerical data needs different tools than categorical data:

  • Categorical (2a): count frequencies in each category
  • Numerical (2b): describe center, spread, and shape of the distribution

Histogram

  • A distribution = values a variable takes and how often.
  • A histogram shows it visually — group values into equal-width intervals (bins), then count how many fall in each.

Dot Plot

A dot plot shows every individual data point — handy when the dataset is small enough to see each value:

Each dot = one Michigan county. For larger datasets, the histogram’s binning becomes more practical.

Center: Mean and Median

Mean: arithmetic average

\bar{x}=\frac{x_1+x_2+\cdots+x_n}{n}

  • Uses every data point

Median: middle value when data are ordered

  • 50% of values above
  • 50% of values below

Center on the Distribution

mean median
78.04 77.85

Mean (red, dashed) sits just above the median (blue, solid) — the distribution leans slightly right.

Compute Center

Five students reported how many hours they volunteered last month:

2, \ 5, \ 8, \ 10, \ 100

(One spent a full week on a campaign event.) Compute — advance for answers:

Question Answer
Mean? (2+5+8+10+100)/5 = 25
Median? the middle value = 8
Which is bigger — and why? mean (the 100 pulls it way up)

Shape: Symmetry and Skewness

Shape describes the overall form of the distribution. Three common shapes:

Symmetric

  • Mirror image around the center
  • Mean ≈ median

Right-skewed

  • Long tail to the right
  • Mean > median

Left-skewed

  • Long tail to the left
  • Mean < median

The MI life expectancy histogram is mildly right-skewed — that’s why the mean came out slightly above the median.

Wayne County Pulls the Mean

Population by Michigan county — a strongly right-skewed distribution:

Mean = 120,028 (red, dashed); median = 37,711 (blue, solid). Wayne County (Detroit, ~1.75M) and a handful of other large counties pull the mean far above the median.

Five-Number Summary

The five-number summary captures a distribution with five values:

  • Minimum, Q1, Median, Q3, Maximum
min Q1 median Q3 max
75.08 77.3 77.85 78.84 81.88

Quartiles are special percentiles — the Pth percentile is the value below which P% of observations fall. Q1 = 25th, the median = Q2 = 50th, Q3 = 75th.

For MI counties: half have life expectancy below 77.8 years, and the middle 50% spans 77.3 to 78.8 years.

Spread: Range, IQR, Standard Deviation

Range = max − min

  • Uses only the two extremes

IQR = Q3 − Q1

  • Spread of the middle 50%

Standard deviation

s = \sqrt{\frac{\sum_{i=1}^n(x_i-\bar{x})^2}{n-1}}

  • Roughly the typical distance from the mean

Spread on the Distribution

range IQR SD
6.8 1.54 1.21

Blue lines mark Q1, median, Q3 (and the band between Q1 and Q3 is the IQR). Red dashed lines mark the mean and ±1 SD. Range = full width of the plot.

Outliers

Outlier: a data point unusually far from the rest. The 1.5×IQR rule flags points more than 1.5×IQR beyond Q1 or Q3.

For MI counties:

  • Lower bound: 77.3 − 1.5×1.54 = 74.99
  • Upper bound: 78.84 + 1.5×1.54 = 81.16

2 outliers — Leelanau County (81.9 years) and Ottawa County (81.3 years), both with unusually high life expectancy.

Find the Outliers

Daily commute time (minutes) for a small class:

min Q1 median Q3 max
5 10 15 22 60

Apply the 1.5 × IQR rule — advance for answers:

Question Answer
IQR? Q3 − Q1 = 22 − 10 = 12
1.5 × IQR? 1.5 × 12 = 18
Lower bound (Q1 − 1.5×IQR)? 10 − 18 = −8
Upper bound (Q3 + 1.5×IQR)? 22 + 18 = 40
Any outliers? yes — the 60-min commute is above the upper bound

Box Plot

A box plot is the five-number summary made visual: box spans Q1 → Q3 (the IQR), line inside the box marks the median, whiskers reach to the min/max excluding outliers, dots are outliers (1.5×IQR rule).

Choosing Appropriate Measures

Center

  • Mean — symmetric data
  • Median — skewed data or outliers

Spread

  • SD — pairs with the mean
  • IQR — pairs with the median

Choose resistant measures (median + IQR) when data are skewed or have outliers.

Reading a Distribution

For this distribution of home sale prices in Ames, Iowa (2,930 homes), decide — advance for answers:

Question Answer
Shape (symmetric / right-skewed / left-skewed)? right-skewed
Mean vs median (>, <, or ≈)? mean > median
Better summary of center — mean or median? median
Better summary of spread — SD or IQR? IQR

References

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