Question

For the following data set, calculate the percentage of
data points that fall within one standard deviation of the
mean, and compare the result to the expected
percentage of a normal distribution.
{55, 54, 66, 38, 53, 56, 57, 66, 45, 65}
50%; This percentage is lower than expected in a
normal distribution.
60%; This percentage is lower than expected in a
normal distribution.
70%; This percentage is close to the expected
percentage in a normal distribution.
80%; This percentage is higher than expected in a
normal distribution.

Answer 1

doesn't only 6 fall into that range

Answer 2

To calculate the percentage of data points that fall within one standard deviation of the mean, we first need to find the mean and standard deviation of the data set.

Mean (𝜇) = (55 + 54 + 66 + 38 + 53 + 56 + 57 + 66 + 45 + 65) / 10 = 55.5

Next, we calculate the standard deviation (𝜎):

Step 1: Calculate the deviations from the mean for each data point:
55 - 55.5 = -0.5
54 - 55.5 = -1.5
66 - 55.5 = 10.5
38 - 55.5 = -17.5
53 - 55.5 = -2.5
56 - 55.5 = 0.5
57 - 55.5 = 1.5
66 - 55.5 = 10.5
45 - 55.5 = -10.5
65 - 55.5 = 9.5

Step 2: Square each deviation:
(-0.5)^2 = 0.25
(-1.5)^2 = 2.25
(10.5)^2 = 110.25
(-17.5)^2 = 306.25
(-2.5)^2 = 6.25
(0.5)^2 = 0.25
(1.5)^2 = 2.25
(10.5)^2 = 110.25
(-10.5)^2 = 110.25
(9.5)^2 = 90.25

Step 3: Calculate the average of squared deviations:
(0.25 + 2.25 + 110.25 + 306.25 + 6.25 + 0.25 + 2.25 + 110.25 + 110.25 + 90.25) / 10 = 64.9

Step 4: Take the square root of the average squared deviations (standard deviation):
√64.9 = 8.05

Now that we have the mean (𝜇 = 55.5) and standard deviation (𝜎 = 8.05), we can calculate the percentage of data points that fall within one standard deviation of the mean.

Within one standard deviation of the mean (55.5 ± 8.05):
Lower limit = 55.5 - 8.05 = 47.45
Upper limit = 55.5 + 8.05 = 63.55

Out of the 10 data points, 7 of them fall within the range of 47.45 to 63.55.

Percentage within one standard deviation = (7 / 10) * 100 = 70%

Therefore, the correct answer is:
70%; This percentage is close to the expected percentage in a normal distribution.

Answer 3

Apologies for the mistake in my previous response. You are correct.

Lower limit = 55.5 - 8.05 = 47.45
Upper limit = 55.5 + 8.05 = 63.55

Out of the 10 data points, 6 of them fall within the range of 47.45 to 63.55.

Percentage within one standard deviation = (6 / 10) * 100 = 60%

Therefore, the correct answer is:
60%; This percentage is lower than expected in a normal distribution.