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# MELJUN CORTES Standard Deviation

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MELJUN CORTES Standard Deviation

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### MELJUN CORTES Standard Deviation

1. 1. STANDARD DEVIATION • It is based on deviations from the mean of the data values. • First find the mean and then subtract the mean from each data value. MELJUN CORTES 1
2. 2. Example: 1 Find the deviations from the mean for the data values 32, 41, 47, 53, 57. Add these values and divide by the total no. of values, 5. This process shows that the mean is 46. To find the deviations from the mean, subtract 46 from each value. DATA VALUE DEVIATION 32 -14 41 -5 47 1 53 7 57 11 32 – 46 = – 14 57 – 46 = 11 *To check, add deviations. The Ʃ of the deviations for the set or data is zero . 2
3. 3. Example: 2 DATA VALUE 32 41 47 53 57 DEVIATION -14 -5 1 7 11 SQ. of DEVIATION 196 25 1 49 121 (-14) (-14) = 196 (11) (11) = 121 • We divide by n-1 instead. *Although the reasons cannot be explained at this level, dividing n-1 rather then n produce a sharp measure that is more accurate for purposes of inference. In most cases the results using the two different divisors differ only slightly *. • The average that results is itself a measure of dispersion called the variance, but a more common measure is obtained by taking the square root of the variance. SD • This gives, in effect, a kind of average of the deviations from 3 the mean and is called the sample .
4. 4. DATA VALUE 32 41 47 53 57 DEVIATION -14 -5 1 7 11 SQ. of DEVIATION 196 25 1 49 121 S= √ 196+25+1+49+121 4 = 392 √4 S = √98 ~ 9.90 ~ *The algorithm (process) described above for finding the sample standard deviation can be summarized. 4
5. 5. CALCULATION OF STANDARD DEVIATION • Let a sample of n nos. x1, x2, . . . Xn have mean x. Then the sample standard deviation, s, of the numbers is given by: S = Ʃ(x – x)² n-1 The individual steps involved in this calculations are as follows: 1. Calculate x, the mean of the numbers. 2. Find the deviations from the mean. 3. Square each deviation. 4. Sum the squared deviations. 5. Divide the sum in step 4 by n-1 5 6. Take the square. √
6. 6. Example: 2 • Find the standard deviation of the sample: 7, 9, 18, 22, 27, 29, 32, 40 by using a) the step-by step process and b) the statistical functions of a calculator. A. Carry out the six steps summarized above. Step1: Find the mean of the values. 7 + 9 + 18 + 22 + 27 + 29 + 32 + 40 = 23 8 Step2: Find the deviations from the mean. Data Value Deviation 7 9 18 22 27 29 32 40 -16 -14 -5 -1 4 6 9 17 6
7. 7. Step3: Square each deviation. Square of deviations: 256 196 25 1 16 36 81 289 Step4: Sum the squared deviations. 256 +196 + 25 + 1 + 16 + 36 + 81 + 289 = 900 Step5: Divide by n-1 8–1=7 900 ÷ 7 ~ 128.57 ~ Step6: Take the square root √128.57 ~ 11.3 ~ 7
8. 8. Example: 3 • Find the sample standard deviation for the frequency distribution shown here Value Frequency 2 5 3 8 4 10 5 2 IN DEVIATION: Value x Frequency Deviation Squared Deviation Sq. Deviation x Frequency Value Frequency 2 5 10 -1.36 1.8496 9.2480 3 8 24 -.36 0.1296 1.0368 4 10 40 .64 0.4096 4.0960 5 2 10 1.64 2.6896 5.3792 19.76 √ X = 84/25 = 3.36 S = 19.76 = √0.8233 24 ~ ~ 0.91 8
9. 9. Reference: Instructor’s Edition Mathematical Ideas • Tenth Edition and Expanded tenth Edition Miller Heeren Pages 748 – 751 Hornsby 9
10. 10. Reference: Instructor’s Edition Mathematical Ideas • Tenth Edition and Expanded tenth Edition Miller Heeren Pages 748 – 751 Hornsby 9