Standard Deviation & the Normal Distribution
Understand standard deviation as a measure of spread and the normal distribution's 68-95-99.7 rule — comparing data sets and finding percentages — with worked SAT and ACT problems.
The Short Version
- Standard deviation measures how spread out data is around the mean — small means tightly clustered.
- Two data sets can share a mean but differ in spread; the one with the larger standard deviation is more variable.
- A normal distribution is the symmetric bell curve where mean = median = mode.
- The 68-95-99.7 rule: about 68%, 95%, and 99.7% of data fall within 1, 2, and 3 standard deviations of the mean.
Averages only tell half the story. Two classes can both average 80 on a test, but if one class scored between 78 and 82 while the other ranged from 50 to 100, those are very different situations. Standard deviation captures that difference — it measures how far, on average, the data sits from the mean. And when data follows the famous bell curve, standard deviation unlocks a set of clean, memorizable percentages.
This guide explains standard deviation, comparing spreads, and the normal distribution's 68-95-99.7 rule, with worked and practice problems matched to real test difficulty at Northside Tutoring.
Why Spread Matters
The SAT and ACT test standard deviation conceptually — they rarely ask you to calculate it by hand. Instead they ask which data set has more spread, or how a change affects the standard deviation, or what percentage falls in a range of a normal distribution. Understanding what spread means is the whole game. It's Algebra II / statistics content beyond the SSAT.
Standard Deviation: Measuring Spread
Standard deviation is, roughly, the typical distance of a data point from the mean. A small standard deviation means the data is bunched tightly around the average; a large one means it's spread out. You won't usually compute it by hand on the test — you'll reason about it.
Comparing Two Data Sets
The comparison rule
Given two data sets, the one whose values cluster closer to the mean has the smaller standard deviation. {2, 2, 3, 3} has a smaller spread than {1, 2, 4, 5}, even if their means match. Look at how far the values stray from the center.
The Normal Distribution
A normal distribution is a symmetric, bell-shaped curve. Most data sits near the center, tapering off equally on both sides. In a perfect normal distribution, the mean, median, and mode are all the same value — the center of the bell.
In a normal distribution, fixed percentages of the data fall within each standard deviation of the mean.
The 68-95-99.7 Rule
Also called the empirical rule, this is the fact the tests want you to know:
| Within… | Contains about… |
|---|---|
| 1 standard deviation of the mean | 68% |
| 2 standard deviations | 95% |
| 3 standard deviations | 99.7% |
Using the Rule
Because the curve is symmetric, you can split these percentages. If 95% lies within 2 standard deviations, then 5% lies outside — split evenly, that's 2.5% in each tail. This kind of reasoning answers most normal-distribution questions without any heavy calculation.
Where You'll See This — Test by Test
No reference sheet covers this. The SAT and ACT test standard deviation as a comparison of spread and the 68-95-99.7 rule for normal data. It's beyond the SSAT.
Digital SAT
Tests comparing standard deviations of two data sets and reasoning about spread — conceptual, not computational.
Explore SAT Tutoring → College AdmissionsACT
Tests the normal distribution and the 68-95-99.7 rule, plus comparing variability.
Explore ACT Tutoring → Independent School AdmissionsSSAT
Not on the SSAT — statistics beyond its scope. Build mean/median fundamentals with earlier prep first.
Explore SSAT Tutoring → K-12 CurriculumAlgebra II
A core statistics topic in Algebra II and AP Statistics.
Explore Algebra Tutoring →Watch the Lesson
Sometimes a diagram needs a voice. In the short video below, one of our Northside tutors walks through the core idea and works through test-style problems in real time.
Standard Deviation — In Plain English
A live walkthrough from our tutoring team.
— Featuring a Northside Tutoring instructor
Worked Example Problems
These problems are calibrated to the difficulty you'll actually see on test day. Try each one before opening the solution.
Which has the larger standard deviation: {10, 10, 10, 10} or {2, 8, 12, 18}?
Show solution
The first set has no spread at all (all values equal), so its standard deviation is 0. The second is spread out.
Test scores are normally distributed with mean 100 and standard deviation 15. About what percent of scores fall between 85 and 115?
Show solution
85 to 115 is within 1 standard deviation (100 ± 15).
By the rule, about 68%.
For the same distribution (mean 100, SD 15), about what percent of scores are above 130?
Show solution
130 is 2 SD above the mean. 95% lie within 2 SD, so 5% lie outside, split into two tails: 2.5% above 130.
In a normal distribution, how do the mean and median compare?
Show solution
A normal distribution is symmetric, so the mean, median, and mode are all equal — at the center.
Two classes average 80. Class A scores range 78–82; Class B ranges 60–100. Which has the larger standard deviation?
Show solution
Class B's scores spread much farther from the mean, so it has the larger standard deviation.
Common Mistakes to Avoid
Three traps that catch students every year
- Confusing spread with center. Standard deviation measures spread, not the average. Two sets with the same mean can have very different SDs.
- Forgetting the curve is symmetric. The 5% outside 2 SD splits into 2.5% per tail — don't report the whole 5% for one side.
- Trying to compute SD by hand. The tests want reasoning about spread and the empirical rule, not a long calculation.
Practice Problems — You Try
Three problems below. Work each before checking the solution.
Heights are normal with mean 65 in and SD 3 in. About what percent are between 59 and 71 inches?
Show solution
59 to 71 is 2 SD on each side (65 ± 6), so about 95%.
Which data set has a standard deviation of 0: {4,4,4,4} or {3,4,5,6}?
Show solution
All identical values means no spread: {4,4,4,4} has SD 0.
Normal data has mean 50, SD 10. About what percent is below 30?
Show solution
30 is 2 SD below the mean. 5% lies outside 2 SD, split into two tails: 2.5% below 30.
The Northside Method — How We Teach This 1-on-1
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