Lecture 5: Descriptive Statistics

Mean, Median, Mode, Standard Deviation

Tom Hanna

2026-02-05

Agenda

  • Lecture today: Descriptive statistics

      - Measures of Central Tendency: Mean, Median, Mode
  - Measures of Dispersion: Variance, Standard Deviation

  • Quiz today:

      - Mean, Median, Mode
  - Bonus points: Variance, SD

  • Wednesday: Discussion

      - Articles 2 and 3
  - Kahan, D. M., & Corbin, J. C. (2016). A note on the perverse effects of actively open-minded thinking on climate-change polarization. Research & Politics, 3(4).
  - Hughes, A. G. (2015). Visualizing inequality: How graphical emphasis shapes public opinion. Research & Politics, 2(4).

Measures of Central Tendency

Measures of central tendency help us:

  • reveal patterns
  • find the typical measurement
  • find the center

Measures of Central Tendency

A few numbers that can summarize the center of measurement

  • Mean

  • Median

  • Mode

Mean

  • Symbol: \(\bar{x}\)
  • Not the middle value
  • Not the most common
  • The center of mass - the sum above equals the sum below
  • Formula is \(\bar{x} = \frac{\sum X_i}{n}\)
  • Read that: The mean of X equals the sum of the observations (i) of X divided by the number (n) of observations.

Example 1:

Find the mean of:

1, 7, 3, 4, 5

  • \(\bar{x} = \frac{\sum X_i}{n}\)

  • \(\bar{x} = \frac{1 + 7 + 3 + 4 + 5}{5}\)

  • \(\bar{x} = 4\)

Example 2:

Find the mean of:

1, 7, 3, 4, 5, 100

  • \(\bar{x} = \frac{\sum X_i}{n}\)

  • \(\bar{x} = \frac{1 + 7 + 3 + 4 + 5 + 100}{6}\)

  • \(\bar{x} = 20\)

Median

  • Midpoint
  • Half observations are greater, half are lower
  • Sort the numbers
  • Then count
  • No formula
  • Even observations - midpoint between middle two (mean of the middle two)

Example 1:

Find the median of:

1, 7, 3, 4, 5

  • Sort the numbers: 1, 3, 4, 5, 7

  • The middle value is 4, so the median is 4.

Example 2:

Find the median of:

1, 7, 3, 4, 5, 100

  • Sort the numbers: 1, 3, 4, 5, 7, 100

  • The middle two values are 4 and 5, so the median is the mean of these two: \(\frac{4 + 5}{2} = 4.5\).

Mode

  • The most common value
  • Can be more than one mode (bimodal, multimodal)
  • Can be no mode (if all values are unique)
  • Not affected by outliers
  • The only measure for nominal data
  • Just count

Example 1:

Find the mode of:

1, 7, 3, 4, 5

  • All values are unique, so there is no mode.

Example 2:

Find the mode of:

1, 7, 3, 4, 5, 7

  • The value 7 appears twice, while all other values appear once, so the mode is 7.

Example 3:

Find the mode of:

1, 7, 3, 4, 5, 7, 3

  • The values 7 and 3 both appear twice, while all other values appear once, so the modes are 7 and 3 (bimodal).

Measures of Dispersion (Variation or Spread)

  • Variance
  • Standard Deviation

Spread

  • We start with the mean
  • Trying to make the picture complete
  • How much do the observations vary around the mean?

Potential measure

  • Just add up the deviations from the mean: \(\sum (X_i - \bar{x})\)
  • But this always equals zero because the mean is the center of mass

Potential measure 2

  • Just add up the absolute value of the deviations from the mean: \(\sum |X_i - \bar{x}|\)
  • Divide this by n to get the average absolute deviation from the mean: \(\frac{\sum |X_i - \bar{x}|}{n}\)
  • This is called the mean absolute deviation (MAD)
  • But this is not used much because it is not mathematically tractable
  • Not useful for statistical inference such as confidence intervals and hypothesis testing

Potential measure 2: Test question

  • Question: There is a much less useful measure of dispersion that is based on the absolute value of deviations from the mean. What is it called?
  • Answer: Mean Absolute Deviation, MAD

Variance

  • What is the other way we can avoid the problem of deviations from the mean summing to zero?
  • Square the deviations from the mean: \(\sum (X_i - \bar{x})^2\)
  • This is called the sum of squared deviations from the mean
  • This number is inflated as the number of observations grows…

Variance (Cont.)

  • Divide by n to get the average squared deviation from the mean:

  • \(\frac{\sum (X_i - \bar{x})^2}{n}\)

  • This is the population variance, \(\sigma^2\) (sigma squared)

  • But we usually don’t have measurements for the entire population

Sample Variance

  • The population variance is systematically too small because the sample mean is closer to the sample observations than the population mean

  • To correct for this bias, we divide by n-1 instead of n to get the sample variance (Bessel’s correction):

  • \(\frac{\sum (X_i - \bar{x})^2}{n-1}\)

  • This is the sample variance, \(s^2\) (s squared)

  • This is an unbiased estimator of the population variance

Parameters and Statistics

  • A parameter is a characteristic of a population (e.g., population mean \(\mu\), population variance \(\sigma^2\))
  • A statistic is a characteristic of a sample (e.g., sample mean \(\bar{x}\), sample variance \(s^2\))
  • We use statistics to estimate parameters

Exam “bonus” question

  • Question: We divide by n-1 instead of n to get an unbiased estimator of the population variance. What is this correction called?
  • Answer: Bessel’s correction

Standard Deviation

  • The variance is in squared units, which can be hard to interpret

  • To make it easier to work with, we want to get back to the original units

  • We take the square root of the variance to get the standard deviation:

  • \(s = \sqrt{\frac{\sum (X_i - \bar{x})^2}{n-1}}\)

or

  • \(s = \sqrt{s^2}\)

  • This is the sample standard deviation, \(s\) (s)

Summary

  • Measures of central tendency: mean, median, mode
  • Measures of dispersion: variance, standard deviation
  • Variance is the average squared deviation from the mean
  • Standard deviation is the square root of the variance
  • The sample variance and standard deviation use n-1 in the denominator to correct for bias (Bessel’s correction)

Practice Application 1

  • If the variance is 100, what is the standard deviation?

  • The standard deviation is the square root of the variance, so \(s = \sqrt{100} = 10\).

Practice Application 2

  • If the standard deviation is 5, what is the variance?

  • The variance is the square of the standard deviation, so \(s^2 = 5^2 = 25\).

Practice Application 3

  • The mean of a sample is 50 and the sum of squared deviations from the mean is 200. If there are 10 observations in the sample, what is the sample variance and standard deviation?

  • The sample variance is calculated as \(s^2 = \frac{\sum (X_i - \bar{x})^2}{n-1} = \frac{200}{10-1} = \frac{200}{9} \approx 22.22\).

  • The sample standard deviation is the square root of the sample variance, so \(s = \sqrt{22.22} \approx 4.71\).