Standard Deviation SAT Notes

Standard deviation measures how far data values typically lie from their mean. A small standard deviation signals tight clustering; a large one signals greater spread. SAT questions often ask for comparisons and transformations rather than long arithmetic.

Spread around the mean

Deviation: For a value $$x_i$$ , its deviation from the mean is $x_i-\bar{x}$ . Deviations may be positive, negative, or zero.

Standard deviation: The nonnegative square root of the average squared distance from the mean for the population-level formula used in this chapter.

Population standard deviation

\[\sigma=\sqrt{\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n}}\]

Subtract the mean, square each deviation, average those squared deviations using $$n$$ , and take the square root.

Deviation calculation for the data set 2, 4, 6
Value $x_i$	Deviation $x_i-\bar{x}$	Squared deviation $(x_i-\bar{x})^2$
$$2$$	$$-2$$	$$4$$
$$4$$	$$0$$	$$0$$
$$6$$	$$2$$	$$4$$

Worked example

Calculate a population standard deviation

Find the population standard deviation of $$2,4,6$$ .

Find the mean
$\bar{x}=(2+4+6)/3=4$ .
Square deviations
The squared deviations are $$4,0,4$$ , totaling $$8$$ .
Average and root
The variance is $$8/3$$ , so $\sigma=\sqrt{8/3}\approx1.63$ .

The population standard deviation is

\sqrt{8/3}

, or about

\(1.63\)

Compare distributions without heavy calculation

Same center, different spread

Smaller standard deviation

Values are more tightly clustered around the mean.

Larger standard deviation

Values are farther from the mean overall.

Transformations of a data set

Effects of common transformations
Transformation	Effect on mean	Effect on standard deviation
Add $$k$$ to every value	Adds $$k$$	No change
Subtract $$k$$ from every value	Subtracts $$k$$	No change
Multiply every value by positive $$k$$	Multiplies by $$k$$	Multiplies by $$k$$
Divide every value by positive $$k$$	Divides by $$k$$	Divides by $$k$$

Worked example

Transform mean and standard deviation

A data set has mean $$18$$ and standard deviation $$4$$ . Each value is transformed by $$y=3x+5$$ .

Scale
Multiplication by $$3$$ changes the mean to $$54$$ and standard deviation to $$12$$ .
Shift
Adding $$5$$ changes the mean to $$59$$ but leaves standard deviation $$12$$ .

The transformed mean is

\(59\)

, and the transformed standard deviation is

\(12\)

Normal distributions and approximate intervals

Normal distribution and standard-deviation intervalsA symmetric bell-shaped curve centered at 70. Guide lines mark one, two, and three standard deviations from the mean.

Approximately 68% lies within 1 standard deviation, 95% within 2, and 99% within 3 at the source-text level.

View chart data

Series	Value	Relative frequency
Normal curve	40	1.11
Normal curve	42.5	2.28
Normal curve	45	4.39
Normal curve	47.5	7.96
Normal curve	50	13.53
Normal curve	52.5	21.63
Normal curve	55	32.47
Normal curve	57.5	45.78
Normal curve	60	60.65
Normal curve	62.5	75.48
Normal curve	65	88.25
Normal curve	67.5	96.92
Normal curve	70	100
Normal curve	72.5	96.92
Normal curve	75	88.25
Normal curve	77.5	75.48
Normal curve	80	60.65
Normal curve	82.5	45.78
Normal curve	85	32.47
Normal curve	87.5	21.63
Normal curve	90	13.53
Normal curve	92.5	7.96
Normal curve	95	4.39
Normal curve	97.5	2.28
Normal curve	100	1.11

Approximate normal-distribution coverage used in this chapter
Interval	Approximate proportion
Within $$1$$ standard deviation of the mean	$68\%$
Within $$2$$ standard deviations of the mean	$95\%$
Within $$3$$ standard deviations of the mean	$99\%$

Worked example

Interpret a normal interval

Scores are approximately normal with mean $$70$$ and standard deviation $$10$$ . About what percent lie from $$60$$ to $$80$$ ?

Locate endpoints
$$60=70-10$$ and $$80=70+10$$ , so the interval is within $$1$$ standard deviation.
Use the rule
Approximately $68\%$ lies within $$1$$ standard deviation.

Approximately

68\%

of scores lie from

\(60\)

\(80\)

Common mistakes and SAT traps

Confusing standard deviation with range; standard deviation uses every observation.
Assuming a larger mean implies a larger standard deviation.
Changing standard deviation after adding the same constant to every value.
Adding a scale factor to standard deviation instead of multiplying by it.
Using the $68\%$ , $95\%$ , and $99\%$ approximations when the distribution is not identified as approximately normal.
Forgetting that one half of a symmetric two-sided region lies on each side of the mean.

Mini check

Check your understanding

A normal distribution has mean $$50$$ and standard deviation $$6$$ . Approximately what percent lies from $$38$$ to $$62$$ ?

$34\%$
$68\%$
$95\%$
$99\%$

Show answer and explanation

Answer: $95\%$

The endpoints are $50\pm2(6)$ , so the interval covers approximately $95\%$ .

Key takeaways

What to remember

Standard deviation measures spread around the mean and is always nonnegative.
Tighter clustering means a smaller standard deviation.
A constant shift leaves standard deviation unchanged; positive scaling multiplies it.
For an approximately normal distribution, use the chapter-level $68\%$ , $95\%$ , and $99\%$ interval approximations.
Check whether a normal-curve region is one-sided or two-sided before selecting a percentage.

Continue learning

Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.

Go to Practice Review Flashcards

Value \(x_i\)	Deviation \(x_i-\bar{x}\)	Squared deviation \((x_i-\bar{x})^2\)
$\(2\)$	$\(-2\)$	$\(4\)$
$\(4\)$	$\(0\)$	$\(0\)$
$\(6\)$	$\(2\)$	$\(4\)$

Transformation	Effect on mean	Effect on standard deviation
Add $\(k\)$ to every value	Adds $\(k\)$	No change
Subtract $\(k\)$ from every value	Subtracts $\(k\)$	No change
Multiply every value by positive $\(k\)$	Multiplies by $\(k\)$	Multiplies by $\(k\)$
Divide every value by positive $\(k\)$	Divides by $\(k\)$	Divides by $\(k\)$

Interval	Approximate proportion
Within $\(1\)$ standard deviation of the mean	$68\%$
Within $\(2\)$ standard deviations of the mean	$95\%$
Within $\(3\)$ standard deviations of the mean	$99\%$