Every distribution we have met so far has its own units - inches, dollars, IQ points, reaction times. That makes them impossible to compare directly. The z-distribution is our universal translator: a way to put any value, from any normal distribution, onto one common ruler. Master this one distribution and the entire inferential half of the book opens up, because every test we run is, underneath, a z-in-disguise.
pnorm, qnorm) instead of a dusty z-table.A z-score answers a single question: how many standard deviations is this value from its mean?
\[z = \frac{x - \bar{x}}{s}\]
That is the whole formula. Subtract the mean (now centered at 0), divide by the standard deviation (now measured in SD-units), and every variable on Earth speaks the same language.
library(tidyverse) # dplyr, ggplot2, purrr, tibble, readr, stringr, forcats
source("_common.R") # book-wide helpers: round2(), fmt_p(), tidy2()
tibble(
test = c("SAT", "ACT"),
score = c(1350, 30),
mean = c(1050, 21),
sd = c(200, 5)
) |>
mutate(z = (score - mean) / sd) |>
round2()| test | score | mean | sd | z |
|---|---|---|---|---|
| SAT | 1350 | 1050 | 200 | 1.5 |
| ACT | 30 | 21 | 5 | 1.8 |
Is a 1350 SAT better than a 30 ACT? On their own scales, who could say. In z-units the question is trivial: the ACT score is further above its mean, so it is the more impressive result. That is the magic of standardizing - incomparable things become comparable.
Standardize a normal variable and you get the standard normal distribution: a bell curve with mean 0 and standard deviation 1. It never changes, which is exactly why it is useful - we can chart it once and reuse it forever.
tibble(z = seq(-4, 4, length.out = 400)) |>
mutate(density = dnorm(z)) |>
ggplot(aes(x = z, y = density)) +
geom_vline(xintercept = -2:2, linetype = "dotted", colour = "grey60") +
geom_line(colour = "steelblue", linewidth = 1) +
labs(x = "z", y = "density") +
theme_book()
The famous 68–95–99.7 rule lives here: about 68% of values fall between \(z=\pm1\), about 95% between \(z=\pm2\), and about 99.7% between \(z=\pm3\). Anything past \(z = \pm2\) is already unusual; past \(\pm3\), rare.
We are about to lean on this one curve for the rest of the book, so it is fair to stop and ask whether we are entitled to. In the best-guess chapter we watched sixteen flat random steps add up into a bell. Adding is one route to the normal. There is a second, and it matters because a great deal of biology and behaviour does not add - it compounds.
Suppose an outcome is produced by a dozen small effects that multiply rather than sum: each one raises or lowers the result by a few percent. Growth works like this. So do many dose-response and learning processes.
set.seed(5)
# 1000 organisms. Growth is 12 small effects MULTIPLYING together - each one
# adds somewhere between 0% and 10% - so nothing here is being summed.
growth <- tibble(organism = 1:1000) |>
mutate(size = map_dbl(organism, \(i) prod(1 + runif(12, min = 0, max = 0.1))))
ggplot(growth, aes(x = size)) +
geom_histogram(aes(y = after_stat(density)), bins = 30,
fill = "seagreen", colour = "white") +
stat_function(fun = dnorm,
args = list(mean = mean(growth$size), sd = sd(growth$size)),
colour = "firebrick", linewidth = 1) +
labs(x = "final size", y = "density") +
theme_book()
A bell again. The reason is quietly elegant: multiplying numbers close to 1 is almost the same as adding the small amounts. \(1.05 \times 1.03 = 1.0815\), and \(1 + 0.05 + 0.03 = 1.08\). Near enough that the addition argument takes over. So small multiplicative effects land in the same place small additive ones do.
That gives us two honest mechanisms rather than a convention:
Being able to defend the normal is not the same as assuming it everywhere, and a psychologist who overclaims here deserves the criticism they will get. The argument fails in specific, recognisable ways:
There is also a distinction worth getting right, because it is the source of a great deal of needless worry. In regression and ANOVA, normality is an assumption about the residuals - what the model failed to explain - not about the raw outcome. A wildly skewed outcome can leave beautifully normal residuals, and a symmetric outcome can leave terrible ones. Look at what the model missed, not at the histogram of \(y\). We show you how in the regression chapter, and what to do when it genuinely fails in Beyond the Normal Curve.
So the honest position, and the one this book takes: the normal distribution earns its place through a mechanism you can demonstrate in three lines, it covers a great deal of what psychological science actually measures, and it has limits you should be able to name. That is a much stronger footing than either assuming it reflexively or abandoning it apologetically.
Older books make you hunt through a printed z-table to convert a z-score into a probability. We have R, so we will never do that. Recall from the distributions chapter that probability is area under the density curve. Two functions give it to you instantly:
pnorm(z) - the area to the left of z (the cumulative probability).qnorm(p) - the inverse: the z-score with area p to its left (a quantile).tibble(
question = c("P(Z < 1): area to the left of z = 1",
"P(Z > 2): the upper tail beyond z = 2",
"P(-2 < Z < 2): the middle ~95%",
"which z cuts off the top 2.5%? (the famous 1.96)"),
answer = c(pnorm(1),
1 - pnorm(2),
pnorm(2) - pnorm(-2),
qnorm(0.975))
) |>
round2()| question | answer |
|---|---|
| P(Z < 1): area to the left of z = 1 | 0.84 |
| P(Z > 2): the upper tail beyond z = 2 | 0.02 |
| P(-2 < Z < 2): the middle ~95% | 0.95 |
| which z cuts off the top 2.5%? (the famous 1.96) | 1.96 |
That last line is worth a star: qnorm(0.975) returns 1.96, the number that shows up in every 95% confidence interval and two-tailed test you will ever run. It is not magic - it is just the z-score with 2.5% of the area beyond it in each tail.
# picture the "unusual" tails beyond +/- 1.96
tibble(z = seq(-4, 4, length.out = 400)) |>
mutate(density = dnorm(z)) |>
ggplot(aes(x = z, y = density)) +
# shade each tail separately, or ggplot joins them across the middle
geom_area(data = \(d) filter(d, abs(z) >= 1.96),
aes(group = z > 0), fill = "red", alpha = 0.3) +
geom_line(colour = "steelblue", linewidth = 1) +
geom_vline(xintercept = c(-1.96, 1.96), colour = "red",
linetype = "dashed", linewidth = 1) +
labs(x = "z", y = "density") +
theme_book()
Turn a z-score into a percentile with pnorm() and you can say exactly where an observation stands in its distribution:
tibble(height = 74, mean = 68, sd = 4) |> # a 74-inch person
mutate(z = (height - mean) / sd,
percentile = pnorm(z) * 100) |>
round2()| height | mean | sd | z | percentile |
|---|---|---|---|---|
| 74 | 68 | 4 | 1.5 | 93.32 |
A z of 1.5 places this person around the 93rd percentile - taller than about 93% of the population. Percentile ranks, growth charts, and standardized-test scores are all pnorm() in another form.
The z-score compares a value to its own sample - to the mean and spread of the data you happened to collect. That is exactly right for asking “how unusual is this person?” But it carries a limitation: because z depends on the sample, the same raw score turns into a different z in a different sample, and z-scores from two different measures are not truly on the same footing.
When the goal is to compare measures - say, is work engagement higher than well-being in this study? - a different ruler is more honest: POMP, the Percent Of Maximum Possible. POMP rescales a score to the range the instrument could produce, from its lowest possible value to its highest:
\[\text{POMP} = 100 \times \frac{X - \text{min}}{\text{max} - \text{min}}\]
A POMP of 0 is the lowest possible score, 100 the highest, and 50 the exact middle of the scale. Crucially, that meaning does not depend on the sample. Two measures with completely different response formats - a 1–5 scale and a 1–7 scale - both land on a common 0–100 scale you can read and compare at a glance:
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 ch05-scores, ch05-pomp. For example, book_data("ch05-scores") in R, Julia, or Python, or !bookdata name = "ch05-scores". in SPSS. Every language reads the same shipped files, so your numbers will match the ones printed here exactly.
pomp <- function(x, lo, hi) 100 * (x - lo) / (hi - lo)
tibble(
measure = c("mood", "vitality"),
score = c(4, 5), # answered on a 1-5 scale and a 1-7 scale
lo = c(1, 1),
hi = c(5, 7)
) |>
mutate(POMP = pomp(score, lo, hi)) |>
round2()| measure | score | lo | hi | POMP |
|---|---|---|---|---|
| mood | 4 | 1 | 5 | 75.00 |
| vitality | 5 | 1 | 7 | 66.67 |
* POMP: 100 * (score - min) / (max - min), using each scale's own
* theoretical minimum and maximum, which travel with the data.
INSERT FILE='data/sim/ch05-pomp.sps'.
COMPUTE POMP = 100 * (score - lo) / (hi - lo).
EXECUTE.
LIST VARIABLES=measure score lo hi POMP. Data List
+--------+--------+--------+--------+-----+
| measure| score | lo | hi | POMP|
+--------+--------+--------+--------+-----+
|mood |4.000000|1.000000|5.000000|75.00|
|vitality|5.000000|1.000000|7.000000|66.67|
+--------+--------+--------+--------+-----+using CSV, DataFrames
df = CSV.read("data/sim/ch05-pomp.csv", DataFrame)
df.POMP = 100 .* (df.score .- df.lo) ./ (df.hi .- df.lo)
df2×5 DataFrame
Row │ measure score lo hi POMP
│ String15 Int64 Int64 Int64 Float64
─────┼────────────────────────────────────────
1 │ mood 4 1 5 75.0
2 │ vitality 5 1 7 66.6667import pandas as pd
df = pd.read_csv("data/sim/ch05-pomp.csv")
def pomp(x, lo, hi):
return 100 * (x - lo) / (hi - lo)
df["POMP"] = pomp(df.score, df.lo, df.hi)
df measure score lo hi POMP
0 mood 4 1 5 75.000000
1 vitality 5 1 7 66.666667On their raw scales, 4 and 5 are hard to compare. In POMP units, the mood score sits at 75% of its possible range and vitality at about 67%, so this person is, if anything, in a slightly better mood than their vitality would suggest. This is the standardization we now use in most of our own papers (PEM): it lets a reader compare outcomes across measures directly, and it has a useful side effect - if a scale’s response format changes partway through a study, the shift in POMP makes the problem hard to miss.
Reach for a z-score to ask how unusual is this value within its distribution? - it locates a person relative to their sample. Reach for POMP to ask how high is this score on what the instrument can measure? - it places a score relative to the possible range, so different measures become comparable. Neither is better; they answer different questions.
Everything inferential rests on one move: standardize, then read the area. When we test a hypothesis, we convert our result to a z-like statistic and ask “how far into the tail is this, if nothing were going on?” When we build a confidence interval, we walk out \(\pm 1.96\) standard errors. The z-distribution is not one more topic - it is the shared machinery of the entire inferential section that follows.
pnorm/qnorm, find (a) the percentile of an IQ of 130, (b) the probability of an IQ above 145, and (c) the IQ that marks the top 1%.scale() and confirm its mean is 0 and sd is 1 (to rounding).We can now describe where a single value stands. Real science, though, asks how two variables move together. The tool that measures that shared movement - and the true engine under correlation and regression - is covariance.