Before you read

Answer first — see what you already know.

Two swim teams practiced for the same event. Team A's times ranged from 70 to 90 seconds with most times near 80. Team B's times ranged from 50 to 100 seconds with times evenly distributed. Which statement is true?

A
Class A has a greater standard deviation
B
Class B has a greater standard deviation
C
Both have the same standard deviation
D
Cannot be determined

Check your answer

Answer:

Before you read

Answer first — see what you already know.

In a medical study, patient blood pressure readings have a mean of 40 mmHg (diastolic) and standard deviation of 5. If every reading is adjusted by adding 10 mmHg (for calibration), what are the new mean and standard deviation?

A
Mean = 50, SD = 5
B
Mean = 50, SD = 15
C
Mean = 40, SD = 15
D
Mean = 50, SD = 50

Check your answer

Answer:

Theory

What Is Standard Deviation?

Standard deviation ( $\sigma$ or $s$ ) measures how spread out data values are from the mean.

- Small SD: data values are clustered near the mean
- Large SD: data values are widely spread

The SAT does not require you to calculate standard deviation by hand. Instead, you need to:
1. Understand what it means
2. Compare SD between data sets
3. Know how changes to data affect SD

Box plot showing the five-number summary

Example 1

Which data set has a greater standard deviation?
Set A: {48, 49, 50, 51, 52}
Set B: {20, 35, 50, 65, 80}

Both have a mean of 50.

Set A: values are within 2 of the mean $\to$ small SD

Set B: values are up to 30 from the mean $\to$ large SD

Set B has a greater standard deviation.

Theory

How Changes Affect Standard Deviation

Adding/subtracting a constant to every value: SD stays the same (shifting doesn't change spread).

Multiplying/dividing every value by a constant $k$ : SD is multiplied/divided by $|k|$ .

Adding an outlier: SD increases.

Removing a value equal to the mean: SD decreases (you removed a 'centered' value).

Example 1

A data set has mean 50 and SD 8. If every value is multiplied by 3, then 10 is added, what are the new mean and SD?

Multiply by 3: Mean $= 150$ , SD $= 24$

Add 10: Mean $= 160$ , SD $= 24$ (adding shifts, doesn't change spread)

New mean = 160, new SD = 24

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