Using the empirical rule to identify values and percentages of a normal distribution

Read the bell-curve picture: how far from the mean do the shaded endpoints sit, measured in standard deviations? That distance picks the empirical-rule percentage (68%, 95%, or 99.7%) and pins down U, V, W.

01 Worked Example Video

Every step of Using the empirical rule to identify values and percentages of a normal distribution explained on video. Pause anywhere; the embed scrolls independently of the page.

YOUTUBE

02 What This Looks Like on ALEKS

ALEKS randomizes the numbers each attempt, but the question shape stays the same. Here are three example versions you might see.

Standardized test scores

Test scores are normal with μ = 76.5, σ = 5.2. The bell curve shows U and W about 3σ from V (almost the entire curve is shaded).

Find W = μ + 3σ.

W =

Adult heights

Adult male heights are normal with μ = 70 in, σ = 3 in. The shaded region is 2σ on each side of V.

Find W = μ + 2σ.

W =

Battery life

Battery life is normal with μ = 12 h, σ = 1.5 h. The shaded region sits 1σ from V on each side (the narrowest common case).

Find W = μ + σ.

W =

Heads up: Your ALEKS version will use different numbers. The numbers in the practice below are different too — that way you're exercising the move, not memorizing one answer.

03 The Rule

V = μ (peak of the curve) U = μ − kσ, W = μ + kσ

k = 1 → 68% · k = 2 → 95% · k = 3 → 99.7% read the picture to pick k

V sits at the peak — that’s just the mean. The picture tells you how many σ out U and W sit. That distance picks the empirical-rule percentage. Then compute U and W exactly with μ ± kσ. See it live: μ = 100, σ = 10, X cutoff at μ + 2σ = 120 in the Normal Distribution Explorer →

Common slips

(1) Read U and W off the picture. The figure is schematic; use μ ± kσ for exact values. (2) Picked the percentage without checking σ distance. The picture pins k first; the percentage just labels which k. (3) Forgot to multiply by k. ±1σ → 68%; ±2σ → 95%; ±3σ → 99.7%. The k = 2 case multiplies σ by 2 before subtracting/adding.

04 Practice — Step by Step

Practice this problem step by step

Scores are normally distributed with μ = 100, σ = 10. On the bell curve, U and W look about 2σ from V on each side. Identify the percentage, V, U, and W.

Step 1

How many σ out?

The picture puts U and W about how many standard deviations from the peak V?

Step 2

Pick the empirical percentage

2σ from the mean corresponds to which empirical-rule percentage?

Step 3

Compute V

V sits at the peak of the bell curve — that’s the mean. What’s V?

V =

Step 4

Compute U

With k = 2, compute U = μ − 2σ = 100 − 2(10).

U =

Step 5

Compute W

Same setup: W = μ + 2σ = 100 + 2(10).

W =

Q28 Q30

Using the empirical rule to identify values and percentages of a normal distribution

You walked the empirical-rule read end to end.