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.
{8, 12, 27, 32, 45, 57, 61, 73, 82, 94}
a. 40%; This percentage is lower than expected in a normal distribution.
b. 50%; This percentage is lower than expected in a normal distribution.
c. 60%; This percentage is lower than expected in a normal distribution.
d. 70%; This percentage is close to the expected percentage in a normal distribution.
To calculate the percentage of data points that fall within one standard deviation of the mean, we first need to calculate the mean and standard deviation of the data set.
The mean is calculated by summing up all the data points and then dividing by the total number of data points:
Mean = (8 + 12 + 27 + 32 + 45 + 57 + 61 + 73 + 82 + 94) / 10 = 481 / 10 = 48.1
Next, we calculate the standard deviation. The standard deviation measures the dispersion or spread of the data points around the mean. The formula to calculate the standard deviation is as follows:
Standard Deviation = sqrt(sum((data point - mean)^2) / n)
where n is the total number of data points.
Standard Deviation = sqrt((8 - 48.1)^2 + (12 - 48.1)^2 + (27 - 48.1)^2 + (32 - 48.1)^2 + (45 - 48.1)^2 + (57 - 48.1)^2 + (61 - 48.1)^2 + (73 - 48.1)^2 + (82 - 48.1)^2 + (94 - 48.1)^2) / 10
= sqrt(17374.9) / 10
≈ sqrt(1737.49) / 10
≈ 41.70 / 10
≈ 4.17
Now we can calculate the percentage of data points that fall within one standard deviation of the mean. In a normal distribution, it is expected that approximately 68% of the data falls within one standard deviation of the mean.
To check which option is correct:
Percentage within one standard deviation = (Number of data points within one standard deviation / Total number of data points) * 100
To determine the number of data points within one standard deviation of the mean, we need to calculate the range of data points that fall within (mean - standard deviation) and (mean + standard deviation).
Mean - Standard Deviation = 48.1 - 4.17 ≈ 43.93
Mean + Standard Deviation = 48.1 + 4.17 ≈ 52.27
Now we count the number of data points in the range (43.93, 52.27):
{8, 12, 27, 32, 45} falls within the range (43.93, 52.27)
Total number of data points in the range = 5
Percentage within one standard deviation = (5 / 10) * 100 = 50%
Since the percentage of data points that fall within one standard deviation of the mean is 50%, option b, "50%; This percentage is lower than expected in a normal distribution," is correct.