Measures of central tendency are a way to summarize a distribution of values with a single value. This single value is the “best guess” for future, unknown values that may be observed. In this section, you will learn about the most common measures of central tendency and how to use them to make sense of your data.
Back in the introduction we asked you to guess a family member’s weight, and claimed your best guess was the arithmetic mean. Here we make good on that claim. There are three classic summaries of “the middle,” and each is the best guess under a different definition of “best.”
The code tabs below assume this chapter’s data is already loaded. Grab the one-file setup for your language from Getting the Book’s Data, run it once, then load what you need by name - this chapter uses ch02-x, ch02-income. For example, book_data("ch02-x") in R, Julia, or Python, or !bookdata name = "ch02-x". in SPSS. Every language reads the same shipped files, so your numbers will match the ones printed here exactly.
library(tidyverse) # dplyr, ggplot2, purrr, tibble, readr, stringr, forcats
source("_common.R") # book-wide helpers: round2(), fmt_p(), tidy2()
x <- c(3, 5, 5, 7, 20) # note the outlier
# R has no built-in mode; count the values and take the most frequent
mode_val <- tibble(x) |> count(x) |> slice_max(n) |> pull(x)
tibble(mean = mean(x), median = median(x), mode = mode_val) |> round2()| mean | median | mode |
|---|---|---|
| 8 | 5 | 5 |
INSERT FILE='data/sim/ch02-x.sps'.
FREQUENCIES VARIABLES=x /STATISTICS=MEAN MEDIAN MODE. Statistics
+---------+--------+
| | x |
+---------+--------+
|N Valid | 5|
| Missing| 0|
+---------+--------+
|Mean | 8.00|
+---------+--------+
|Median | 5.00|
+---------+--------+
|Mode |5.000000|
+---------+--------+
x
+---------------+---------+-------+-------------+------------------+
| |Frequency|Percent|Valid Percent|Cumulative Percent|
+---------------+---------+-------+-------------+------------------+
|Valid 3.000000 | 1| 20.0%| 20.0%| 20.0%|
| 5.000000 | 2| 40.0%| 40.0%| 60.0%|
| 7.000000 | 1| 20.0%| 20.0%| 80.0%|
| 20.000000| 1| 20.0%| 20.0%| 100.0%|
+---------------+---------+-------+-------------+------------------+
|Total | 5| 100.0%| | |
+---------------+---------+-------+-------------+------------------+using CSV, DataFrames, Statistics, StatsBase
df = CSV.read("data/sim/ch02-x.csv", DataFrame)
mean(df.x)
median(df.x)
mode(df.x)8.0
5.0
5import pandas as pd
df = pd.read_csv("data/sim/ch02-x.csv")
df.x.mean()
df.x.median()
df.x.mode()[0] # mode() returns every tie; take the firstnp.float64(8.0)
np.float64(5.0)
np.int64(5)Here is the deep idea that ties this chapter to the entire rest of the book. Suppose you must pick a single number \(c\) to guess every value, and you will be penalized by the squared error of each guess. Which \(c\) minimizes your total penalty? Let’s just try every candidate and see:
# try every candidate guess and record its total squared error
guesses <- tibble(guess = seq(0, 20, by = 0.1)) |>
mutate(sse = map_dbl(guess, \(g) sum((x - g)^2)))
best <- guesses |> slice_min(sse) |> pull(guess)
ggplot(guesses, aes(x = guess, y = sse)) +
geom_line(colour = "steelblue", linewidth = 1) +
geom_vline(xintercept = mean(x), colour = "red",
linetype = "dashed", linewidth = 1) +
labs(x = "guess (c)", y = "sum of squared errors") +
theme_book()
tibble(best_guess = best, the_mean = mean(x)) |> round2()| best_guess | the_mean |
|---|---|
| 8 | 8 |
The valley of that curve sits exactly at the mean. That is not a coincidence - it is the reason the mean matters. When we do regression later and “minimize the sum of squared residuals,” we are doing this same thing in more dimensions. The mean is least-squares estimation with no predictors.
The median has its own claim to fame: it minimizes the sum of absolute errors, \(\sum |x - c|\). Different penalty, different winner. And the mode simply asks “what shows up most?” - the only one of the three that makes sense for a categorical variable like ice-cream flavor.
We keep drawing bell curves, and you are entitled to ask why. Psychologists get told, often by people outside the field, that assuming a normal distribution is naive - that real behaviour is messy and the bell is a convenient fiction. That criticism deserves a real answer rather than a defensive one, and the answer is a demonstration you can run in three lines.
Put a thousand people on the centre line of a football field. Each one takes sixteen steps. Every step is a shove of random size, somewhere between a metre left and a metre right, and every distance in that range is equally likely. Nothing here is bell-shaped: a single step is perfectly flat, with no preference for small shoves over large ones.
Now look at where everybody ends up.
set.seed(2)
# 1000 people, 16 steps each. One step is FLAT - any distance from -1 to 1 is
# equally likely - so nothing about the ingredients is bell-shaped.
walk <- tibble(person = 1:1000) |>
mutate(position = map_dbl(person, \(i) sum(runif(16, min = -1, max = 1))))
ggplot(walk, aes(x = position)) +
geom_histogram(aes(y = after_stat(density)), bins = 30,
fill = "steelblue", colour = "white") +
# the normal curve implied by the spread we actually got
stat_function(fun = dnorm, args = list(mean = 0, sd = sd(walk$position)),
colour = "firebrick", linewidth = 1) +
labs(x = "final position on the field", y = "density") +
theme_book()
A bell, and a close one. We did not assume it, choose it, or fit it - it fell out of adding.
The mechanism is worth saying in plain words, because once you see it you will recognise it everywhere. To finish far to the right, a person needs nearly every one of their sixteen steps to push right. There is essentially one way to do that. To finish near the middle, the pushes need only roughly cancel, and there are an enormous number of ways for that to happen. Extremes require agreement; middles do not. Add up enough independent nudges and that lopsidedness in the counting is the bell.
Here is why this matters for psychological science in particular. Look at what our measurements actually are:
These are not variables we hope are normal. They are sums, and sums of many small independent influences are how bells get made. The normal distribution is not an assumption imposed on psychology from outside; for a great deal of what we measure, it is the arithmetic of measurement showing through.
That is a genuine defence, and it has genuine limits, which we take up honestly when the normal becomes our working tool in the z-distribution chapter. For now, notice what the demonstration bought us: a reason the mean sits at the centre of so many distributions, and therefore a reason it so often earns its place as the best guess.
Look again at that outlier (20). Squaring errors makes the mean very sensitive to extreme values; one billionaire walks into the room and the “average” net worth becomes absurd. The median shrugs it off:
income <- c(30, 35, 40, 42, 45, 48, 5000) # thousands; one very rich person
tibble(mean = mean(income), median = median(income)) |> round2()| mean | median |
|---|---|
| 748.57 | 42 |
INSERT FILE='data/sim/ch02-income.sps'.
FREQUENCIES VARIABLES=income /STATISTICS=MEAN MEDIAN. Statistics
+---------+------+
| |income|
+---------+------+
|N Valid | 7|
| Missing| 0|
+---------+------+
|Mean |748.57|
+---------+------+
|Median | 42.00|
+---------+------+
income
+-----------------+---------+-------+-------------+------------------+
| |Frequency|Percent|Valid Percent|Cumulative Percent|
+-----------------+---------+-------+-------------+------------------+
|Valid 30.000000 | 1| 14.3%| 14.3%| 14.3%|
| 35.000000 | 1| 14.3%| 14.3%| 28.6%|
| 40.000000 | 1| 14.3%| 14.3%| 42.9%|
| 42.000000 | 1| 14.3%| 14.3%| 57.1%|
| 45.000000 | 1| 14.3%| 14.3%| 71.4%|
| 48.000000 | 1| 14.3%| 14.3%| 85.7%|
| 5000.000000| 1| 14.3%| 14.3%| 100.0%|
+-----------------+---------+-------+-------------+------------------+
|Total | 7| 100.0%| | |
+-----------------+---------+-------+-------------+------------------+using CSV, DataFrames, Statistics
df = CSV.read("data/sim/ch02-income.csv", DataFrame)
mean(df.income)
median(df.income)748.5714285714286
42.0import pandas as pd
df = pd.read_csv("data/sim/ch02-income.csv")
df.income.mean()
df.income.median()np.float64(748.5714285714286)
np.float64(42.0)The median (about 42) describes a typical person; the mean (about 749) describes nobody in the room. This is why you should reach for the median whenever a distribution is skewed or outlier-ridden, and the mean when it is roughly symmetric.
Median or mean isn’t the only choice. Robust estimators - the trimmed mean (drop a fixed fraction of each tail, average the rest) and the MAD (a median-based cousin of the SD) - keep most of the mean’s efficiency while resisting the outliers that wreck it. The mean’s breakdown point is essentially 0% (one bad value can move it anywhere); the median’s is 50%. We develop this fully in Beyond the Normal Curve.
There is no universally “correct” measure of center. There is only the one that answers your question honestly.
y <- rexp(200). Compute its mean and median. Which is larger, and why?y, then make a second plot of \(\sum|y - c|\) and confirm its minimum sits at the median.y (say, c(y, 500)) and recompute both. Which summary barely moved? State the lesson in one sentence.A center is only half the story. Two distributions can share the same mean and be wildly different - one tightly packed, one spread all over. That spread is our uncertainty, and it is the single most important quantity in statistics. On to dispersion.