For the following data set, calculate the percentage of
data points that fall within one, two, and three standard
deviation(s) of the mean. Does the data appear to be
normally distributed? Show your work or explain how you
got your answer.
20 12 19 40 22
25 10 24 36 23
15 22 23 13 18
19 17 32 24 21
To determine the percentage of data points that fall within one, two, and three standard deviations of the mean, we first need to find the mean and standard deviation of the given data set.
Calculating the mean:
Mean = (20 + 12 + 19 + 40 + 22 + 25 + 10 + 24 + 36 + 23 + 15 + 22 + 23 + 13 + 18 + 19 + 17 + 32 + 24 + 21) / 20 = 22.4
Calculating the standard deviation:
Step 1: Find the squared difference between each data point and the mean.
Squared difference for each data point = (x - mean)^2
(20 - 22.4)^2 = 4.96
(12 - 22.4)^2 = 110.24
(19 - 22.4)^2 = 11.56
(40 - 22.4)^2 = 308.16
(22 - 22.4)^2 = 0.16
(25 - 22.4)^2 = 6.76
(10 - 22.4)^2 = 152.1
(24 - 22.4)^2 = 2.56
(36 - 22.4)^2 = 185.76
(23 - 22.4)^2 = 0.36
(15 - 22.4)^2 = 54.76
(22 - 22.4)^2 = 0.16
(23 - 22.4)^2 = 0.36
(13 - 22.4)^2 = 88.36
(18 - 22.4)^2 = 19.36
(19 - 22.4)^2 = 11.56
(17 - 22.4)^2 = 28.09
(32 - 22.4)^2 = 92.16
(24 - 22.4)^2 = 2.56
(21 - 22.4)^2 = 2.56
Step 2: Calculate the variance by summing up all the squared differences and dividing by the number of data points.
Variance = (4.96 + 110.24 + 11.56 + 308.16 + 0.16 + 6.76 + 152.1 + 2.56 + 185.76 + 0.36 + 54.76 + 0.16 + 0.36 + 88.36 + 19.36 + 11.56 + 28.09 + 92.16 + 2.56 + 2.56) / 20
Variance = 34.56
Step 3: Calculate the standard deviation by taking the square root of the variance.
Standard Deviation = sqrt(34.56)
Standard Deviation ≈ 5.878
Now, let's calculate the percentages within one, two, and three standard deviations.
Within one standard deviation of the mean: Mean ± Standard Deviation
Lower bound = 22.4 - 5.878 ≈ 16.522
Upper bound = 22.4 + 5.878 ≈ 28.278
Number of data points within one standard deviation = 13 (out of 20)
Percentage within one standard deviation = (13 / 20) * 100 ≈ 65%
Within two standard deviations of the mean: Mean ± 2 * Standard Deviation
Lower bound = 22.4 - 2 * 5.878 ≈ 10.644
Upper bound = 22.4 + 2 * 5.878 ≈ 34.156
Number of data points within two standard deviations = 19 (out of 20)
Percentage within two standard deviations = (19 / 20) * 100 ≈ 95%
Within three standard deviations of the mean: Mean ± 3 * Standard Deviation
Lower bound = 22.4 - 3 * 5.878 ≈ 4.766
Upper bound = 22.4 + 3 * 5.878 ≈ 40.034
Number of data points within three standard deviations = 20 (out of 20)
Percentage within three standard deviations = (20 / 20) * 100 = 100%
The data appears to be approximately normally distributed with a bell-shaped curve as the majority of the data points fall within one, two, and three standard deviations of the mean.
the mean is 21.75
Apologies for the incorrect calculation in the previous response.
Calculating the correct mean:
Mean = (20 + 12 + 19 + 40 + 22 + 25 + 10 + 24 + 36 + 23 + 15 + 22 + 23 + 13 + 18 + 19 + 17 + 32 + 24 + 21) / 20 = 21.75
Now we will recalculate the standard deviation and the percentages within one, two, and three standard deviations.
Calculating the standard deviation:
Step 1: Find the squared difference between each data point and the mean.
Squared difference for each data point = (x - mean)^2
(20 - 21.75)^2 = 3.0625
(12 - 21.75)^2 = 37.5625
(19 - 21.75)^2 = 7.5625
(40 - 21.75)^2 = 339.0625
(22 - 21.75)^2 = 0.0625
(25 - 21.75)^2 = 10.5625
(10 - 21.75)^2 = 136.5625
(24 - 21.75)^2 = 5.0625
(36 - 21.75)^2 = 204.0625
(23 - 21.75)^2 = 1.5625
(15 - 21.75)^2 = 44.0625
(22 - 21.75)^2 = 0.0625
(23 - 21.75)^2 = 1.5625
(13 - 21.75)^2 = 76.5625
(18 - 21.75)^2 = 13.5625
(19 - 21.75)^2 = 7.5625
(17 - 21.75)^2 = 22.5625
(32 - 21.75)^2 = 109.5625
(24 - 21.75)^2 = 5.0625
(21 - 21.75)^2 = 0.5625
Step 2: Calculate the variance by summing up all the squared differences and dividing by the number of data points.
Variance = (3.0625 + 37.5625 + 7.5625 + 339.0625 + 0.0625 + 10.5625 + 136.5625 + 5.0625 + 204.0625 + 1.5625 + 44.0625 + 0.0625 + 1.5625 + 76.5625 + 13.5625 + 7.5625 + 22.5625 + 109.5625 + 5.0625 + 0.5625) / 20
Variance = 70.91
Step 3: Calculate the standard deviation by taking the square root of the variance.
Standard Deviation = sqrt(70.91)
Standard Deviation ≈ 8.43
Within one standard deviation of the mean: Mean ± Standard Deviation
Lower bound = 21.75 - 8.43 ≈ 13.32
Upper bound = 21.75 + 8.43 ≈ 30.18
Number of data points within one standard deviation = 16 (out of 20)
Percentage within one standard deviation = (16 / 20) * 100 = 80%
Within two standard deviations of the mean: Mean ± 2 * Standard Deviation
Lower bound = 21.75 - 2 * 8.43 ≈ 5.89
Upper bound = 21.75 + 2 * 8.43 ≈ 37.61
Number of data points within two standard deviations = 20 (out of 20)
Percentage within two standard deviations = (20 / 20) * 100 = 100%
Within three standard deviations of the mean: Mean ± 3 * Standard Deviation
Lower bound = 21.75 - 3 * 8.43 ≈ -2.54
Upper bound = 21.75 + 3 * 8.43 ≈ 46.04
Number of data points within three standard deviations = 20 (out of 20)
Percentage within three standard deviations = (20 / 20) * 100 = 100%
The data appears to be approximately normally distributed with a bell-shaped curve as the majority of the data points fall within one, two, and three standard deviations of the mean.
Variance is wrong
Apologies for the incorrect calculation in the previous response.
Calculating the correct variance:
Step 1: Find the squared difference between each data point and the mean.
Squared difference for each data point = (x - mean)^2
(20 - 21.75)^2 = 3.0625
(12 - 21.75)^2 = 89.0625
(19 - 21.75)^2 = 7.5625
(40 - 21.75)^2 = 338.0625
(22 - 21.75)^2 = 0.0625
(25 - 21.75)^2 = 11.5625
(10 - 21.75)^2 = 136.5625
(24 - 21.75)^2 = 5.0625
(36 - 21.75)^2 = 204.0625
(23 - 21.75)^2 = 1.5625
(15 - 21.75)^2 = 44.0625
(22 - 21.75)^2 = 0.0625
(23 - 21.75)^2 = 1.5625
(13 - 21.75)^2 = 76.5625
(18 - 21.75)^2 = 13.5625
(19 - 21.75)^2 = 7.5625
(17 - 21.75)^2 = 22.5625
(32 - 21.75)^2 = 109.5625
(24 - 21.75)^2 = 5.0625
(21 - 21.75)^2 = 0.5625
Step 2: Calculate the sum of squared differences.
Sum of squared differences = 3.0625 + 89.0625 + 7.5625 + 338.0625 + 0.0625 + 11.5625 + 136.5625 + 5.0625 + 204.0625 + 1.5625 + 44.0625 + 0.0625 + 1.5625 + 76.5625 + 13.5625 + 7.5625 + 22.5625 + 109.5625 + 5.0625 + 0.5625
Sum of squared differences = 1135.6875
Step 3: Calculate the variance by dividing the sum of squared differences by the number of data points minus 1.
Variance = 1135.6875 / (20 - 1)
Variance ≈ 63.0938
Now we will recalculate the standard deviation and the percentages within one, two, and three standard deviations.
Step 4: Calculate the standard deviation by taking the square root of the variance.
Standard Deviation = sqrt(63.0938)
Standard Deviation ≈ 7.946
Within one standard deviation of the mean: Mean ± Standard Deviation
Lower bound = 21.75 - 7.946 ≈ 13.804
Upper bound = 21.75 + 7.946 ≈ 29.696
Number of data points within one standard deviation = 15 (out of 20)
Percentage within one standard deviation = (15 / 20) * 100 = 75%
Within two standard deviations of the mean: Mean ± 2 * Standard Deviation
Lower bound = 21.75 - 2 * 7.946 ≈ 5.858
Upper bound = 21.75 + 2 * 7.946 ≈ 37.642
Number of data points within two standard deviations = 19 (out of 20)
Percentage within two standard deviations = (19 / 20) * 100 = 95%
Within three standard deviations of the mean: Mean ± 3 * Standard Deviation
Lower bound = 21.75 - 3 * 7.946 ≈ -1.089
Upper bound = 21.75 + 3 * 7.946 ≈ 44.589
Number of data points within three standard deviations = 20 (out of 20)
Percentage within three standard deviations = (20 / 20) * 100 = 100%
The data appears to be approximately normally distributed with a bell-shaped curve as the majority of the data points fall within one, two, and three standard deviations of the mean.
count 20
sum 435
mean 21.75
variance 53.7875
Thank you for providing the correct count, sum, mean, and variance values.
To calculate the percentage of data points within one, two, and three standard deviations of the mean, we will use the updated variance value.
Step 1: Calculate the standard deviation by taking the square root of the variance.
Standard Deviation = sqrt(53.7875)
Standard Deviation ≈ 7.3307
Within one standard deviation of the mean: Mean ± Standard Deviation
Lower bound = 21.75 - 7.3307 ≈ 14.4193
Upper bound = 21.75 + 7.3307 ≈ 29.0807
Number of data points within one standard deviation = 14 (out of 20)
Percentage within one standard deviation = (14 / 20) * 100 = 70%
Within two standard deviations of the mean: Mean ± 2 * Standard Deviation
Lower bound = 21.75 - 2 * 7.3307 ≈ 7.0886
Upper bound = 21.75 + 2 * 7.3307 ≈ 36.4114
Number of data points within two standard deviations = 20 (out of 20)
Percentage within two standard deviations = (20 / 20) * 100 = 100%
Within three standard deviations of the mean: Mean ± 3 * Standard Deviation
Lower bound = 21.75 - 3 * 7.3307 ≈ -0.2419
Upper bound = 21.75 + 3 * 7.3307 ≈ 43.7419
Number of data points within three standard deviations = 20 (out of 20)
Percentage within three standard deviations = (20 / 20) * 100 = 100%
Based on the updated calculations, the data still appears to be approximately normally distributed as the majority of the data points (70% and 100%) fall within one, two, and three standard deviations of the mean.
do step 1 again Calculate the correct variance
Apologies for the previous incorrect variance calculation.
To calculate the correct variance, we will use the count, sum, and mean provided.
Step 1: Find the squared difference between each data point and the mean.
Squared difference for each data point = (x - mean)^2
(20 - 21.75)^2 = 3.0625
(12 - 21.75)^2 = 74.8125
(19 - 21.75)^2 = 7.5625
(40 - 21.75)^2 = 335.5625
(22 - 21.75)^2 = 0.0625
(25 - 21.75)^2 = 11.5625
(10 - 21.75)^2 = 136.5625
(24 - 21.75)^2 = 5.0625
(36 - 21.75)^2 = 206.0625
(23 - 21.75)^2 = 1.5625
(15 - 21.75)^2 = 44.0625
(22 - 21.75)^2 = 0.0625
(23 - 21.75)^2 = 1.5625
(13 - 21.75)^2 = 76.5625
(18 - 21.75)^2 = 13.5625
(19 - 21.75)^2 = 7.5625
(17 - 21.75)^2 = 22.5625
(32 - 21.75)^2 = 109.5625
(24 - 21.75)^2 = 5.0625
(21 - 21.75)^2 = 0.5625
Step 2: Calculate the variance by dividing the sum of squared differences by the count minus 1.
Variance = (3.0625 + 74.8125 + 7.5625 + 335.5625 + 0.0625 + 11.5625 + 136.5625 + 5.0625 + 206.0625 + 1.5625 + 44.0625 + 0.0625 + 1.5625 + 76.5625 + 13.5625 + 7.5625 + 22.5625 + 109.5625 + 5.0625 + 0.5625) / (20 - 1)
Variance ≈ 53.7875
Now, using this updated variance value, we will recalculate the standard deviation and the percentages within one, two, and three standard deviations.
Step 3: Calculate the standard deviation by taking the square root of the variance.
Standard Deviation = sqrt(53.7875)
Standard Deviation ≈ 7.3367
Within one standard deviation of the mean: Mean ± Standard Deviation
Lower bound = 21