The annual sales, in millions of dollars, for 28 toy department stores is listed.
38 35 43 27 24 40 38
21 34 27 35 36 24 33
27 33 44 45 23 33 33
24 33 27 32 35 29 31
A)Calculate the mean,(u), and standard deviationof the data. Answer to the nearest tenth.
(My answer): U=32.3 SD=6.3
B) Calculate the percent sales, to the nearest whole number
i)Within one SD of the mean.
?????????????????????
ii)Within two SD of the mean.
??????????????????????????
On your standard deviation, I did sigma n-1 and got 6.45. I am not certain what you did.
b. I think it means count the values which are within +- one SD. Count means add them, then divide by the total sales.
For a i used my calc. The 1-Var Stats
To calculate the mean, follow these steps:
Step 1: Begin by adding up all the values in the dataset. The sum of the given values is:
38 + 35 + 43 + 27 + 24 + 40 + 38 + 21 + 34 + 27 + 35 + 36 + 24 + 33 + 27 + 33 + 44 + 45 + 23 + 33 + 33 + 24 + 33 + 27 + 32 + 35 + 29 + 31 = 887
Step 2: Divide the sum by the number of data points. In this case, there are 28 stores, so:
Mean (u) = 887 / 28 = 31.68 (rounded to the nearest tenth)
To calculate the standard deviation, follow these steps:
Step 1: Find the difference between each data point and the mean.
For example, for the first data point (38), the difference is 38 - 31.68 = 6.32.
Similarly, calculate the difference for each data point.
Step 2: Square each of the differences obtained in the previous step.
For example, for the first data point (38), the squared difference is (6.32)^2 = 39.94.
Step 3: Calculate the average of all the squared differences obtained.
Add up all the squared differences:
(6.32)^2 + (3.32)^2 + (11.32)^2 + (4.68)^2 + (6.68)^2 + (8.32)^2 + (6.32)^2 + (9.68)^2 + (2.68)^2 + (4.68)^2 + (3.32)^2 + (4.32)^2 + (6.68)^2 + (1.68)^2 + (3.32)^2 + (1.32)^2 + (12.32)^2 + (13.32)^2 + (7.68)^2 + (1.32)^2 + (1.32)^2 + (6.68)^2 + (1.32)^2 + (11.32)^2 + (0.68)^2 + (3.32)^2 + (9.68)^2 + (5.32)^2 + (0.32)^2 = 1224.16
Divide the sum by the number of data points (28):
Standard deviation (SD) = sqrt(1224.16 / 28) = 6.33 (rounded to the nearest tenth)
To calculate the percent sales within one standard deviation of the mean, follow these steps:
Step 1: Determine the upper and lower limits within one standard deviation.
Upper limit = mean + standard deviation = 31.68 + 6.33 = 37.01 (rounded to two decimal places)
Lower limit = mean - standard deviation = 31.68 - 6.33 = 25.35 (rounded to two decimal places)
Step 2: Count the number of sales values that fall within this range.
In the given data, there are 19 sales values within one standard deviation of the mean.
To calculate the percent, divide the count by the total number of sales values and multiply by 100:
Percent Sales Within One SD = (19 / 28) * 100 = 67.9% (rounded to the nearest whole number)
To calculate the percent sales within two standard deviations of the mean, follow the same steps for calculating one standard deviation, but use the upper and lower limits for two standard deviations:
Upper limit = mean + (2 x standard deviation) = 31.68 + (2 x 6.33) = 44.34 (rounded to two decimal places)
Lower limit = mean - (2 x standard deviation) = 31.68 - (2 x 6.33) = 19.02 (rounded to two decimal places)
Now count the number of sales values that fall within this range. In the given data, there are 25 sales values within two standard deviations of the mean.
To calculate the percent, divide the count by the total number of sales values and multiply by 100:
Percent Sales Within Two SD = (25 / 28) * 100 = 89.3% (rounded to the nearest whole number)
Therefore, the answers to part A) are:
- Mean (u) = 31.7 (rounded to the nearest tenth)
- Standard Deviation (SD) = 6.3 (rounded to the nearest tenth)
The answers to part B) are:
- i) Percent Sales within one SD = 68% (rounded to the nearest whole number)
- ii) Percent Sales within two SD = 89% (rounded to the nearest whole number)