1. Standard Deviation!

2. MENWOMEN 0.901.50 2.003.00 1.403.00 2.002.50 3.00 2.003.00 4.00 3.00 3.702.00 MEN: _____ WOMEN:______ Let's say we randomly select 9 men and 9 women and ask their GPAs and get these data: MEAN GPA FOR: 2.44 2.78 Can we conclude that the average GPA of ALL women at Skyline is higher than the average of ALL men?

3. Standard Deviation The standard deviation is a measure the degree each data point in the set of data points varies (or deviates) from the mean The wider the spread of scores, the larger the standard deviation.

4. Standard Deviation For data that has a normal distribution, 68% of the data lies within one standard deviation of the mean. One SD

5. MEN: _____ WOMEN:______ How to calculate the standard deviation: 1. Calculate the mean (M) of a set of data MEAN GPA FOR: 2.44 2.78

6. MENWOMEN 0.90–2.44=1.50–2.78= 2.00–2.44= -0.44 3.00–2.78= 0.22 1.40–2.44= -1.04 3.00–2.78= 0.22 2.00–2.44= -0.44 2.50–2.78= -0.28 3.00–2.44= 0.56 3.00–2.78= 0.22 2.00–2.44= -0.44 3.00–2.78= 0.22 3.00–2.44= 0.56 4.00–2.78= 1.22 4.00–2.44= 1.56 3.00–2.78= 0.22 3.70–2.44= 1.26 2.00–2.78= -0.78 How to calculate the standard deviation: 2. Subtract the mean from each point of data to determine (X-M)

7. How to calculate the standard deviation: 3. Square each of the resulting numbers to determine (X-M) 2 MENWOMEN (-1.54) 2 = (-0.44 ) 2 = ( -1.04 ) 2 = ( 0.22 ) 2 = ( -0.44 ) 2 = ( 0.56 ) 2 = ( -0.44 ) 2 = ( 0.56 ) 2 = ( 1.56 ) 2 = ( 0.22 ) 2 = ( 1.22 ) 2 = ( 0.22 ) 2 = ( 1.26 ) 2 = 2.37 (-1.28 ) 2 = 1.64 0.19 1.08 0.05 0.19 ( -0.28 ) 2 = 0.08 0.31 0.19 0.31 2.43 0.05 1.49 0.05 1.59 ( -0.78 ) 2 = 0.61

8. How to calculate the standard deviation: 4. Add the values from the previous step together to get ∑(X-M) 2 MEN 2.37 0.19 1.08 0.19 0.31 0.19 0.31 2.43 1.59 8.68 WOMEN 1.64 0.05 0.08 0.05 1.49 0.05 0.61 4.06

9. How to calculate the standard deviation: 5. Calculate (n-1) by subtracting 1 from your sample size. Your sample size is the total number of data points you collected. N-1 for men = 9-1 = 8 N-1 for women = 9-1 = 8

10. How to calculate the standard deviation: 6. Divide the answer from ∑(X-M) 2 by the answer from (n-1) to find ∑(X-M) 2 n-1 Men Women 8.68 / 8 = 1.09 4.06 / 8 = 0.51

11. How to calculate the standard deviation: 7. Calculate the square root of your previous answer to determine the standard deviation Men √1.09 = Women √0.51 = 1.04 0.71

12. This means that… In men, 68% of all students sampled have a GPA that falls within 1.04 grade points of the mean. – MEAN PLUS 1.04 = 3.49 – MEAN MINUS 1.04 = 1.40 In women, 68% of all students sampled have a GPA that falls within 0.71 grade points of the mean. – MEAN PLUS 0.71 = 3.49 – MEAN MINUS 0.71 = 2.07 68% of men have a GPA between these values 68% of women have a GPA between these values

13. GPA 3.00 2.50 2.00 1.50 1.00 0.50 0.00 So, are women smarter? Graph results to find out: Mean GPAof Men and Women at Skyline 4.00 3.50 MENWOMEN + / - 1 SD MEAN GPA Gender

14. GPA 4.00 3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.00 MENWOMEN Gender to the SD we GPAof Men that the So, are women smarter? Graph results Since find out: bars overlap, Mean conclude and Women at Skyline GPA’s are probably NOT significantly different, and that the difference in the means is most likely due to chance in the sampling population.

15. Since the SD bars DO NOT overlap, we conclude that the populations are probably significantly different, and that the difference in the means is most likely due to something MORE THAN CHANCE.

16. Using Tools to Calculate Standard Deviation Excel Type the values you are trying to find the mean for in a column. You can label the column, but you don’t have to. Determine which box you want the mean to appear in. In the example, I want the mean to appear in box A12. In that box, type: =AVERAGE(A2:A11) and then hit enter. Basically you are telling Excel to average boxes A2 through A11. Determine which box you want the standard deviation to appear in. In the example, I want the standard deviation to appear in box A13. In that box, type: =STDEV(A2:A11) and then hit enter. You are giving Excel the box labels for the data for which you want to find the standard deviation.

17. TI-83 First we have to enter the data. Hit the STAT button and you will see the options EDIT, CALC and TESTS atop the screen. Use the left and right arrows (if necessary) to move the cursor to EDIT, then select 1:Edit... Now you will see a table with the headings L 1 and L 2. Enter the values under L 1 (if you want to clear pre-existing data first, move the cursor to the top of the column, hit CLEAR and then ENTER.) Once all the data is entered, go back to the STAT menu, but this time move the cursor to CALC instead of EDIT. Once you're in the CALC menu, select 1-Var Stats, then hit ENTER. The calculator will display the x-mean, some other stuff, and then the standard deviation (s x ). Note that s x is what we called s.d. in class; the calculator refers to it as s x. This is followed by something called sigma x (which is what you would get as standard deviation if you had used n instead of n-1), and finally the sample size.