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}
(1 point)
Responses
40%; This percentage is lower than expected in a normal distribution.
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.
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.
The mean is calculated by summing all the numbers in the data set and dividing by the total number of data points:
Mean = (8 + 12 + 27 + 32 + 45 + 57 + 61 + 73 + 82 + 94) / 10 = 51.1
Next, we calculate the standard deviation. To do this, we need to find the squared difference between each data point and the mean, sum them up, divide by the total number of data points, and finally take the square root of the result.
Standard deviation = sqrt(((8 - 51.1)^2 + (12 - 51.1)^2 + (27 - 51.1)^2 + (32 - 51.1)^2 + (45 - 51.1)^2 + (57 - 51.1)^2 + (61 - 51.1)^2 + (73 - 51.1)^2 + (82 - 51.1)^2 + (94 - 51.1)^2) / 10) ≈ 27.03
Now, we can calculate the percentage of data points within one standard deviation of the mean. In a normal distribution, approximately 68% of the data would fall within one standard deviation of the mean.
To find the number of data points within one standard deviation, we need to count how many data points are within the range mean - standard deviation to mean + standard deviation.
In this case, the range is 51.1 - 27.03 to 51.1 + 27.03, which is approximately 24.07 to 78.17.
Looking at the data set, the data points that fall within this range are 27, 32, 45, 57, 61, and 73. So there are 6 data points within one standard deviation.
To find the percentage, we divide the number of data points within one standard deviation by the total number of data points and multiply by 100:
Percentage = (6 / 10) * 100 = 60%
Comparing this result to the expected percentage of a normal distribution, which is approximately 68%, we can see that the calculated percentage is lower. Therefore, the correct answer is:
60%; This percentage is lower than expected in a normal distribution.