Question

The tables below give the distribution of high temperatures in degrees Fahrenheit (${}^{o}F$) for City A and City B over the same 21 days in March.
City A

Temperature (${}^{o}F$)	Frequency
80	3
79	14
78	2
77	1
76	1

City B

Temperature (${}^{o}F$)	Frequency
80	6
79	3
78	2
77	4
76	6

Which of the following is true about the data shown for these 21 days?

• A. The standard deviation of temperatures in City A is larger.
• B. The standard deviation of temperatures in City B is larger.
• C. The standard deviation of temperatures in City A is the same as that of City B.
• D. The standard deviation of temperatures in these cities cannot be calculated with the data provided.

Answer 1

Answer: (B)

The standard deviation of temperatures in City B is larger.

For City $B$

$x$	$f$	$fx$	$x^2$	$f{x}^2$
80 79 78 77 76	6 3 2 4 6	480 237 156 308 456	6400 6241 6084 5929 5776	38400 18723 12168 23716 34656

$\sum fx=1637,\sum f{x}^2=127663,\sum f=21$

$\overline{x}=Mean=\frac{\sum fx}{\sum t}=77.95$

${\sigma }^2=\frac{\sum f{x}^2}{\sum f}-(\overline{x}{)}^2=\frac{127663}{21}-(77.95{)}^2=2.987$

$\sigma =1.728$

For City $A$

$x$	$f$	$fx$	$x^2$	$f{x}^2$
80 79 78 77 76	3 14 02 01 01	240 1106 156 77 76	6400 6241 6084 5929 5776	19200 87374 12168 5929 5776

$\sum fx=1655,\sum f{x}^2=130447,\sum f=21$

$\overline{x}=Mean=\frac{\sum fx}{\sum t}=78.80$

${\sigma }^2=\frac{\sum f{x}^2}{\sum f}-(\overline{x}{)}^2=\frac{130447}{21}-(78.80{)}^2=2.321$

$\sigma =1.523$

So, we can say that standard deviation of City $B$ is larger.