Introduction to Statistics · Chapter 5 · The Normal Distribution

The Standard Normal Distribution

Every normal distribution can be converted into one universal ruler — the z-score — so values from entirely different scales can be compared directly.

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The Standard Normal Distribution · bookSHelf Intro Stats§5.1

Learning objectives — by the end of this section you will be able to

Objectives

Explain what the standard normal distribution is, and why its mean is zero and its standard deviation is one definition
Calculate a z-score from a value, a mean, and a standard deviation, and read its sign and size as a position §5.1.1
Recover the original value $x$ from a z-score using $x = \mu + z\sigma$ inverse formula
Compare two values from differently scaled normal distributions by comparing their z-scores portability
Apply the Empirical Rule to say what fraction of the data lies within one, two, and three standard deviations of the mean §5.1.2

The Standard Normal Distribution · bookSHelf Intro Stats§5.1

§5.1.1 — reading the formula, piece by piece

Two Moves, One Ruler

The numerator $x - \mu$ measures how far a value sits from the center, in the original units — positive above the mean, negative below. The denominator $\sigma$ asks how that gap compares to the distribution's own typical spread.

Subtracting recenters the distribution at zero; dividing rescales it so one unit of spread becomes one unit of measurement. Together, that is what makes the score portable — a count of standard deviations, not a count of centimeters.

The Standard Normal Distribution · bookSHelf Intro Stats§5.1

§5.1.1 — the target of every standardization

The Standard Normal Distribution

Definition 5.1.1 — Standard Normal Distribution

The standard normal distribution is the normal distribution with mean zero and standard deviation one, written $Z \sim N(0, 1)$ . Its values are called standardized values, or z-scores.

The transformation

z = \frac{x - \mu}{\sigma}

turns $X \sim N(\mu, \sigma)$ into $Z \sim N(0, 1)$ .

Definition 5.1.1: Standardizing changes the ruler under the curve, not the curve.

The Standard Normal Distribution · bookSHelf Intro Stats§5.1

Insight — one ruler for every measurement

A change of units, not a new idea

A z-score is a change of units, the way converting to meters lets you compare a Canadian road sign with an American one. Once both values are in standard deviations, the original scales stop mattering and you can put them side by side.

The Standard Normal Distribution · bookSHelf Intro Stats§5.1

§5.1.1 — the working formula

The z-Score

Definition 5.1.2 — z-Score

If $X \sim N(\mu, \sigma)$ , then the z-score of a value $x$ is

z = \frac{x - \mu}{\sigma}

Definition 5.1.2: The z-score is the number of standard-deviation steps from the mean to the value.

Rearranging the same equation gives you the trip back: $x = \mu + z\sigma$ — the form to reach for whenever a problem hands you a standardized score and asks what it was before standardizing.

The Standard Normal Distribution · bookSHelf Intro Stats§5.1

Your turn — substitute and simplify

Try It Now 5.1.1

Try It Now 5.1.1 — a single z-score

What is the z-score of $x$ , when $x = 1$ and $X \sim N(12, 3)$ ?

Answer: $z = \frac{1 - 12}{3} = \frac{-11}{3} \approx -3.67$ The value $x = 1$ sits about 3.67 standard deviations below the mean of 12 — far out in the left tail.

The Standard Normal Distribution · bookSHelf Intro Stats§5.1

Worked example — the sign says which side

Example 5.1.1: Reading a z-Score Both Ways

Worked Example 5.1.1 — $X \sim N(5, 6)$

Find and interpret the z-score for $x = 17$ , and then for $x = 1$ .

$x = 17$ : $z = \frac{17 - 5}{6} = 2$ two standard deviations to the right of the mean 5. Check: $\mu + z\sigma = 5 + (2)(6) = 17$ .

$x = 1$ : $z = \frac{1 - 5}{6} \approx -0.67$ about 0.67 standard deviations to the left of the mean 5.

The Standard Normal Distribution · bookSHelf Intro Stats§5.1

Your turn — fill in the blanks

Try It Now 5.1.2

Try It Now 5.1.2 — Camila's points per game

Camila averages 16 points a game with a standard deviation of four, so $X \sim N(16, 4)$ . She scores ten points in a game: $z = -1.5$ . This tells you $x = 10$ is ____ standard deviations to the ____ of the mean ____.

Answer: $x = 10$ is 1.5 standard deviations to the left of the mean 16. $z = \frac{10 - 16}{4} = \frac{-6}{4} = -1.5$

The Standard Normal Distribution · bookSHelf Intro Stats§5.1

Worked example — a gain is a negative loss

Example 5.1.2: Weight Loss in Pounds

Worked Example 5.1.2 — $X \sim N(5, 2)$ , weight lost in a month

a. Mai lost ten pounds: $z = \frac{10-5}{2} = 2.5$ — 2.5 sd to the right of the mean five.

b. Her brother gained three pounds, so $x = -3$ : $z = \frac{-3-5}{2} = -4$ — four sd to the left of the mean.

c. $Y \sim N(2,1)$ , $y = 4$ : $z = \frac{4-2}{1} = 2$ — the same $z = 2$ as $x = 17$ in Example 5.1.1, each two sd right of its own mean.

The Standard Normal Distribution · bookSHelf Intro Stats§5.1

Your turn — both directions

Try It Now 5.1.3

Try It Now 5.1.3 — Chilean male heights, $X \sim N(170, 6.28)$

a. A male was 176 cm tall — find $z$ . b. A male has $z = -2$ — find his height.

a. $z = \frac{176-170}{6.28} \approx 0.96$ about 0.96 sd to the right of 170 cm.

b. $x = 170 + (-2)(6.28) = 157.44 \text{ cm}$ two sd to the left of the mean.

The Standard Normal Distribution · bookSHelf Intro Stats§5.1

Worked example — same distribution, both directions

Example 5.1.3: Heights of Chilean Males, 2009 to 2010

Worked Example 5.1.3 — $X \sim N(170, 6.28)$ cm

a. A male was 168 cm tall — find $z$ . b. A male has $z = 1.27$ — find his height.

a. $z = \frac{168-170}{6.28} \approx -0.32$ 0.32 sd to the left of the mean 170.

b. $x = 170 + (1.27)(6.28) \approx 177.98 \text{ cm}$ 1.27 sd to the right of the mean.

The Standard Normal Distribution · bookSHelf Intro Stats§5.1

Your turn — comparing two scores

Try It Now 5.1.4

Try It Now 5.1.4 — SAT verbal, $X \sim N(496, 114)$

Find the z-scores for $x_1 = 325$ and $x_2 = 366.21$ . Which score sits further from the mean?

$z_1 = \frac{325-496}{114} = -1.5 \qquad z_2 = \frac{366.21-496}{114} \approx -1.14$ Answer: both are below the mean, and 325 sits deeper in the left tail than 366.21 does.

The Standard Normal Distribution · bookSHelf Intro Stats§5.1

Worked example — two different populations, one shared answer

Example 5.1.4: Comparing Two Chilean Height Distributions

Worked Example 5.1.4 — $Y \sim N(172.36, 6.34)$ [1984–85], $X \sim N(170, 6.28)$ [2009–10]

Find the z-scores for $x = 160.58$ cm and $y = 162.85$ cm.

$\frac{160.58-170}{6.28} = -1.5 \qquad \frac{162.85-172.36}{6.34} = -1.5$ Answer: the raw heights differ by more than 2 cm, from two distributions with different centers and spreads — but both standardize to $z = -1.5$ . Relative to the population each was drawn from, the two males were equally short.

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Try it in rāSHio — proving a shared z-score

Take it on the panel, not on faith

Example 5.1.4 claims the two heights sit at the same place in their own distributions. Choose Distributions → Normal, pick Left, and enter mean 170, standard deviation 6.28, value 160.58; then run it again with 172.36, 6.34, and 162.85 — the same area comes back both times, which is what a shared z-score of $-1.5$ actually means.

The Standard Normal Distribution · bookSHelf Intro Stats§5.1

§5.1.1 — the panel, in motion

Reading a Tail Area in rāSHio

Figure 5.1.1: Reading a left-tail area in rāSHio: Distributions → Normal, demonstrated on $X \sim N(100, 15)$ with a cutoff at 115 — the panel's twin axes label that point in x-units and in z at once.

The Standard Normal Distribution · bookSHelf Intro Stats§5.1

§5.1.2 — reading the curve

Why the Percentages Aren't Evenly Spaced

The curve is tallest in the middle, so most of the area is accounted for early: going from one standard deviation to two only buys another 27 percentage points, and two to three buys about 4.7 more.

That is what makes a z-score past 3 genuinely remarkable. The exact areas are 68.27%, 95.45%, and 99.73% — a problem quoting those longer figures is still asking about the same three bands.

The Standard Normal Distribution · bookSHelf Intro Stats§5.1

§5.1.2 — three bands, every normal distribution

The Empirical Rule

Definition 5.1.3 — The Empirical Rule

about 68% of the $x$ values lie within $1\sigma$ of the mean $\mu$ ;
about 95% of the $x$ values lie within $2\sigma$ of the mean $\mu$ ;
about 99.7% of the $x$ values lie within $3\sigma$ of the mean $\mu$ .

Figure 5.1.2: The normal curve divided at $\mu$ and at each standard deviation out to $3\sigma$ .

Also known as the 68–95–99.7 rule. The z-scores at each edge are exactly $\pm1$ , $\pm2$ , and $\pm3$ .

The Standard Normal Distribution · bookSHelf Intro Stats§5.1

Context Pause — where "1 in 20" comes from

Two standard deviations, one rule of thumb

The 95% band in the rule is why researchers so often treat two standard deviations as the line between ordinary and surprising. Anything past $2\sigma$ happens to about one observation in twenty — rare enough to be worth a second look.

The Standard Normal Distribution · bookSHelf Intro Stats§5.1

Your turn — the 68% band

Try It Now 5.1.5

Try It Now 5.1.5 — $X \sim N(25, 5)$

Between what values of $x$ do 68% of the values lie?

Answer: $25 - (1)(5) = 20 \qquad 25 + (1)(5) = 30$ about 68% of the values lie between $x = 20$ and $x = 30$ .

The Standard Normal Distribution · bookSHelf Intro Stats§5.1

Worked example — all three bands at once

Example 5.1.5: Applying the Rule to a Mean of 50

Worked Example 5.1.5 — $X \sim N(50, 6)$

Band	Lower	Upper	z-scores
68%	44	56	−1, +1
95%	38	62	−2, +2
99.7%	32	68	−3, +3

The Standard Normal Distribution · bookSHelf Intro Stats§5.1

Try it in rāSHio — the exact areas behind the rule

Rounded bands, exact areas

The three bands in Example 5.1.5 are rounded areas, and rāSHio will hand you the exact ones. Choose Distributions → Normal, pick Between, set mean 50 and standard deviation 6, then read the area for 44–56, then 38–62, then 32–68 — 0.6827, 0.9545, and 0.9973, the numbers the 68-95-99.7 rule rounds off.

The Standard Normal Distribution · bookSHelf Intro Stats§5.1

Your turn — an entrance exam

Try It Now 5.1.6

Try It Now 5.1.6 — $\mu = 52$ , $\sigma = 11$ points

About 68%, 95%, and 99.7% of scores lie between what two values, respectively?

41–63 ( $z=\mp1$ ) · 30–74 ( $z=\mp2$ ) · 19–85 ( $z=\mp3$ )

The Standard Normal Distribution · bookSHelf Intro Stats§5.1

Worked example — the rule, in centimeters

Example 5.1.6: The Empirical Rule on Chilean Heights, 1984 to 1985

Worked Example 5.1.6 — $Y \sim N(172.36, 6.34)$ cm

Band	Lower (cm)	Upper (cm)	z-scores
68%	166.02	178.70	−1, +1
95%	159.68	185.04	−2, +2
99.7%	153.34	191.38	−3, +3

The Standard Normal Distribution · bookSHelf Intro Stats§5.1

Key Terminology — the vocabulary this section put to work

standard normal distribution — the normal distribution with mean zero and standard deviation one, written $Z \sim N(0, 1)$ .

z-score — the number of standard deviations a value lies above or below the mean of its own distribution, $z = (x - \mu)/\sigma$ .

standardized value — a value re-expressed as a z-score, so that distributions on different scales can be compared directly.

Empirical Rule — for a normal distribution, about 68%, 95%, and 99.7% of the data lie within one, two, and three standard deviations of the mean; also called the 68-95-99.7 rule.

The Standard Normal Distribution · bookSHelf Intro Stats§5.1

The headline result

The Empirical Rule: 68 – 95 – 99.7

Within one, two, and three standard deviations of the mean, in any normal distribution — once every value is standardized to the same $Z \sim N(0,1)$ ruler.

The same z-score formula, $z = (x-\mu)/\sigma$ , is what makes the rule apply everywhere at once: standardize a value first, and its position under these three bands never depends on the original units.

† These are the rounded approximations; the exact areas are 68.27%, 95.45%, and 99.73% — both forms describe the same three bands.

5.1

The Standard Normal Distribution · bookSHelf Intro Stats§5.1

§5.1 — conclusions

What §5.1 Leaves You With

The one idea

$z = (x-\mu)/\sigma$ converts any normal value into a universal ruler measured in standard deviations, so values from differently scaled distributions become directly comparable — same z-score, same relative position.

When to reach for more

The Empirical Rule only covers whole standard deviations — 1, 2, and 3 $\sigma$ . For anything in between, or a probability more exact than three rounded bands, use rāSHio's Distributions → Normal panel instead.

Next: §5.2 — Using the Normal Distribution, where the z-score becomes the tool for reading probabilities off any cutoff. Back to start.