Math 115: Introduction to Statistics

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

Your turn again. Here’s a five-number summary for daily commute times in a small class — minimum five minutes, Q one ten, median fifteen, Q three twenty-two, maximum sixty. Use the one and a half IQR rule to find the IQR, the lower bound, the upper bound, and any outliers. Advance the slide when you’re ready to check your answers. The IQR is Q three minus Q one — twenty-two minus ten — which is twelve. One and a half times the IQR is eighteen. The lower bound is Q one minus that — ten minus eighteen — which is negative eight. The upper bound is Q three plus that — twenty-two plus eighteen — equals forty. The minimum, five, is well above the lower bound, so no low-side outliers. The maximum, sixty, is above forty, so it’s a high-side outlier — the long-commute student stands out from the rest.

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

Introduction to Modern Statistics (2e) textbook by Mine Çetinkaya-Rundel and Johanna Hardin
Sections 5.2–5.6