A botanist compares the heights, in centimeters, of two different varieties of sunflower, … — SAT

A botanist compares the heights, in centimeters, of two different varieties of sunflower, Variety A and Variety B, grown under identical greenhouse conditions. The table below shows the distribution of the heights for $6$ sunflowers of each variety.

Height (cm)	Variety A frequency	Variety B frequency
$110$	$0$	$1$
$115$	$0$	$1$
$120$	$2$	$1$
$121$	$2$	$0$
$122$	$2$	$1$
$127$	$0$	$1$
$132$	$0$	$1$

Which of the following statements correctly compares the standard deviations of the heights of Variety A and Variety B?

A
The standard deviation of Variety A is greater than the standard deviation of Variety B.
The standard deviation of Variety B is greater than the standard deviation of Variety A.Answer
C
The standard deviations of the heights of Variety A and Variety B are equal.
D
The standard deviation of Variety B is equal to the mean height of Variety B.

Answer

The standard deviation of Variety B is greater than the standard deviation of Variety A.

The correct statement is that the standard deviation of Variety B is greater than that of Variety A. The standard deviation measures how spread out the values in a dataset are from the mean. For Variety A, all data points are very close to the mean height of

121\text{ cm}

. For Variety B, the heights are spread much further from the mean, ranging from

110\text{ cm}

132\text{ cm}

. Since Variety B's data points show greater variability and are more spread out, the standard deviation of Variety B must be greater than the standard deviation of Variety A.

Step-by-Step Solution

Understand the concept of standard deviation.

Standard deviation measures the variability or spread of data points around their mean.

This sets up the conceptual framework for comparing the two varieties without needing complex calculations.

Analyze the distribution and spread of Variety A.

The heights of Variety A are clustered closely, with all values (

120

121

, and

122

) within

1\text{ cm}

of the mean height of

121\text{ cm}

This indicates that Variety A has a relatively low standard deviation due to its high concentration of data around the mean.

Analyze the distribution and spread of Variety B.

The heights of Variety B are spread out, ranging from

110\text{ cm}

132\text{ cm}

This demonstrates that Variety B has much higher variability around the mean than Variety A.

Compare the standard deviations based on the analyzed spreads.

Since Variety B is more spread out than Variety A, the standard deviation of Variety B must be greater than the standard deviation of Variety A.

This leads directly to the correct statement.

Key Concept

Standard deviation is a measure of how spread out the values in a dataset are from the mean.

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