Question

the table below shows the scores on a math test.
david\t75\taaron\t75
ashley\t60\tgina\t75
andrew\t75\tkei\t95
monica\t100\tanna\t60
lucas\t75\tjennifer\t65

find the standard deviation. round to the nearest tenth when necessary.
a. 157.25\tc. 9.1
b. 12.5\td. 12.3

please select the best answer from the choices provided

Explanation:

Step 1: Find the mean ($\mu$)

First, list all the scores: 75, 75, 60, 75, 75, 95, 100, 60, 75, 65.
The number of scores ($n$) is 10.
Sum of scores: $75 + 75 + 60 + 75 + 75 + 95 + 100 + 60 + 75 + 65 = 750$
Mean $\mu = \frac{750}{10} = 75$

Step 2: Calculate each deviation from the mean and square it

• For 75: $(75 - 75)^2 = 0$ (there are 5 scores of 75, so total for these: $5 \times 0 = 0$)
• For 60: $(60 - 75)^2 = (-15)^2 = 225$ (there are 2 scores of 60, so total: $2 \times 225 = 450$)
• For 95: $(95 - 75)^2 = 20^2 = 400$ (1 score, total: 400)
• For 100: $(100 - 75)^2 = 25^2 = 625$ (1 score, total: 625)
• For 65: $(65 - 75)^2 = (-10)^2 = 100$ (1 score, total: 100)

Step 3: Find the sum of squared deviations

Sum of squared deviations: $0 + 450 + 400 + 625 + 100 = 1575$

Step 4: Calculate variance ($\sigma^2$)

Variance $\sigma^2 = \frac{\text{Sum of squared deviations}}{n} = \frac{1575}{10} = 157.25$

Step 5: Calculate standard deviation ($\sigma$)

Standard deviation $\sigma = \sqrt{157.25} \approx 12.5$ (rounded to the nearest tenth)

Answer:

b. 12.5