Unformatted text preview: Calculating the Mean Calculating the Variance and Standard Deviation
2 REMEMBER: The frequency is the NOTE (1) When calculating the variance, we multiply the value (data ‐ mean)
NOTE: (1) Wh
ti th
lti th
t number of times a specific value times the frequency just like we did when we calculated the mean. occurred; e.g., we know from the (2) Since the problem asks us to round to 2 DP, we will cautiously round only table below that 3 students our second‐to‐last calulcation to 2+2 (or 4) DP, but all other calculations will received a 70 on their test. exact be done with "exact" values. TSCR FREQ 70 3 TSCR x FREQ = …
70 x 3 = 210 2 TSCR
70 (TSCR ‐ MEAN) 385.322359396433… = 385.3224 3 385.3224 x 3 = 1155.9672 92.7297668038408… = 92.7298 5 92.7298 x 5 = 463.6490 2 0.137174211248288… = 0.1372 11 0.1372 x 11 = 1.5092 2 107.544581618656… = 107.5446 6 107.5446 x 6 = 645.2676 2 414.951989026063… = 414.9520 2 414.952 x 2 = 829.9040 60 x 5 = 300 60 ( 60 1360/27 ) = ‐ 1360/27 50 11 50 x 11 = 550 50 ( 50 ‐ 1360/27 ) = 30 2 40 x 6 = 240 30 x 2 = 27 60
1360 Mean = 1360/27 = 50.370370370… ≈ 50.4 NOTE: Use "1360/27" for the
mean when calculating the
variance to avoid rounding error 40
30 2 (TSCR ‐ MEAN) x FREQ = … 2 5 6 FREQ 2 ( 70 ‐ 1360/27 ) = 60 40 = … ( 40 ‐ 1360/27 ) = ( 30 ‐ 1360/27 ) = 27 Variance = 3096.2970/27 ≈ 114.68 and SD = SQRT(114.68) ≈ 10.71 3096.297 Finding the Variance and Standard Deviation The procedure for finding the standard deviation that is presented in the textbook i.e., the green box below works well for finding the standard deviation of a list of numbers; however, the procedure does not lend itself to data presented in a frequency table. The good news is if we make a very minor change in the procedure above, it will work well for a list of numbers and numbers presented in a frequency table! Let’s look at the textbook procedure and the improved procedure side‐by‐side… The Textbook Procedure The Improved Procedure
Step 1 Determine the mean of the set of numbers. Step 1 Determine the mean of the set of numbers.
Step 2 Subtract the mean from each number.
Step 2 Subtract the mean from each number.
Step 3 Square each of these differences. Step 3 Square each of these differences. Step 4 Find the sum of the squares of these Step 4 Find the mean of the numbers from Step 3.
differences. NOTE: If the original set of numbers came from a frequency table, find the mean of these numbers Step 5 Divide this sum by the number of pieces of using the same method used in Step 1.
data. The result here is the variance. The result here is the variance. Step 6 Take the square root of the variance.
Step 5 Take the square root of the variance. The result here is the standard deviation. The result here is the standard deviation. Since Steps 4 and 5 in the textbook procedure is nothing more than finding the mean of the numbers from Step 3, the improved procedure is really no different than the textbook procedure. The only real difference is the improved procedure anticipates the possibility that your data may have come from a frequency table and warns you to manipulate the numbers from Step 3 accordingly. The next few pages include examples of both lists of numbers and numbers presented in a frequency table… Find the mean, variance, and standard deviation of the following: 79, 90, 95, 95, 96 ① Determine the mean. (Note: So our calculations are not too unwieldy, when finding the variance and standard deviation in this course, you may round the mean to the nearest whole number.) ② Subtract the mean from each number. ③ Square each of these differences. ④ Find the mean of the numbers found in Step 3; the result is the variance. (Note : Round the variance to the nearest tenth.) ⑤ Take the square root of the variance; the result is the standard deviation. (Note : Round the standard deviation to the nearest tenth.) Find the mean, variance, and standard deviation of the following: 4, 7, 10, 7, 5, 2, 7 ① Determine the mean. (Note: So our calculations are not too unwieldy, when finding the variance and standard deviation in this course, you may round the mean to the nearest whole number.) ② Subtract the mean from each number. ③ Square each of these differences. ④ Find the mean of the numbers found in Step 3; the result is the variance. (Note : Round the variance to the nearest tenth.) ⑤ Take the square root of the variance; the result is the standard deviation. (Note : Round the standard deviation to the nearest tenth.) Find the mean, variance, and standard deviation of the following test results. ① Test Score Frequency 90 80 70 60 50 3 5 12 7 4 Determine the mean. (Note: So our calculations are not too unwieldy, when finding the variance and standard deviation in this course, you may round the mean to the nearest whole number.) Remember : Since the data is presented in a frequency table, we must use the following method for finding the mean! ② Subtract the mean from each number. ③ Square each of these differences. ④ Find the mean of the numbers found in Step 3; the result is the variance. (Note : Round the variance to the nearest tenth.) Remember: Since the original data was presented in a frequency table, we must use the same method used in Step 1 for finding this mean, too! ⑤ Take the square root of the variance; the result is the standard deviation. (Note : Round the standard deviation to the nearest tenth.) Find the mean, variance, and standard deviation of the following test results. ① Test Score Frequency 90 80 70 60 50 4 6 8 8 5 Determine the mean. (Note: So our calculations are not too unwieldy, when finding the variance and standard deviation in this course, you may round the mean to the nearest whole number.) Remember : Since the data is presented in a frequency table, we must use the following method for finding the mean! ② Subtract the mean from each number. ③ Square each of these differences. ④ Find the mean of the numbers found in Step 3; the result is the variance. (Note : Round the variance to the nearest tenth.) Remember: Since the original data was presented in a frequency table, we must use the same method used in Step 1 for finding this mean, too! ⑤ Take the square root of the variance; the result is the standard deviation. (Note : Round the standard deviation to the nearest tenth.) ...
View
Full
Document
This note was uploaded on 02/29/2012 for the course MATH 120 taught by Professor Adjunct during the Fall '11 term at Northern Virginia Community College.
 Fall '11
 Adjunct
 Math, Variance

Click to edit the document details