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### Standard Deviation and Variance

• 1.
• 2. OBJECTIVES The learners are expected to: a. Calculate the Standard Deviation of a given set of data. b. Calculate the Variance of a given set of data.
• 3. STANDARD DEVIATION  is a special form of average deviation from the mean.  is the positive square root of the arithmetic mean of the squared deviations from the mean of the distribution.
• 4. STANDARD DEVIATION  is considered as the most reliable measure of variability.  is affected by the individual values or items in the distribution.
• 5. Standard Deviation for Ungrouped Data
• 6. How to Calculate the Standard Deviation for Ungrouped Data 1. Find the Mean. 2. Calculate the difference between each score and the mean. 3. Square the difference between each score and the mean.
• 7. How to Calculate the Standard Deviation for Ungrouped Data 4. Add up all the squares of the difference between each score and the mean. 5. Divide the obtained sum by n – 1. 6. Extract the positive square root of the obtained quotient.
• 8. Find the Standard Deviation 35 73 35 11 35 49 35 35 35 15 35 27 210 210 Mean= 35 Mean= 35
• 9. Find the Standard Deviation x x-ẋ (x-ẋ)2 x x-ẋ (x-ẋ)2 35 0 0 73 38 1444 35 0 0 11 -24 576 35 0 0 49 14 196 35 0 0 35 0 0 35 0 0 15 -20 400 35 0 0 27 -8 64 ∑(x-ẋ)2 0 ∑(x-ẋ)2 2680
• 10. Find the Standard Deviation
• 11. How to Calculate the Standard Deviation for Grouped Data 1. Calculate the mean. 2. Get the deviations by finding the difference of each midpoint from the mean. 3. Square the deviations and find its summation. 4. Substitute in the formula.
• 12. Find the Standard Deviation Class F Midpoint FMp _ _ _ _ Limits (2) (3) (4) X Mp - X (Mp-X)2 f( Mp-X)2 (1) 28-29 4 28.5 114.0 20.14 8.36 69.89 279.56 26-27 9 26.5 238.5 20.14 6.36 40.45 364.05 24-25 12 24.5 294.0 20.14 4.36 19.01 228.12 22-23 10 22.5 225.0 20.14 2.36 5.57 55.70 20-21 17 20.5 348.5 20.14 0.36 0.13 2.21 18-19 20 18.5 370.0 20.14 -1.64 2.69 53.80 16-17 14 16.5 231.0 20.14 -3.64 13.25 185.50 14-15 9 14.5 130.5 20.14 -5.64 31.81 286.29 12-13 5 12.5 62.5 20.14 -7.64 58.37 291.85 N= ∑fMp= ∑(Mp-X)2= 100 2,014.0 1,747.08
• 13. Find the Standard Deviation
• 14. Characteristics of the Standard Deviation 1. The standard deviation is affected by the value of every observation. 2. The process of squaring the deviations before adding avoids the algebraic fallacy of disregarding the signs.
• 15. Characteristics of the Standard Deviation 3. It has a definite mathematical meaning and is perfectly adapted to algebraic treatment. 4. It is, in general, less affected by fluctuations of sampling than the other measures of dispersion.
• 16. Characteristics of the Standard Deviation 5. The standard deviation is the unit customarily used in defining areas under the normal curve of error. It has, thus, great practical utility in sampling and statistical inference.
• 17. VARIANCE  is the square of the standard deviation. In short, having obtained the value of the standard deviation, you can already determine the value of the variance.
• 18. VARIANCE It follows then that similar process will be observed in calculating both standard deviation and variance. It is only the square root symbol that makes standard deviation different from variance.
• 20. How to Calculate the Variance for Ungrouped Data 1. Find the Mean. 2. Calculate the difference between each score and the mean. 3. Square the difference between each score and the mean.
• 21. How to Calculate the Variance for Ungrouped Data 4. Add up all the squares of the difference between each score and the mean. 5. Divide the obtained sum by n – 1.
• 22. Find the Variance 35 73 35 11 35 49 35 35 35 15 35 27 210 210 Mean= 35 Mean= 35
• 23. Find the Variance x x-ẋ (x-ẋ)2 x x-ẋ (x-ẋ)2 35 0 0 73 38 1444 35 0 0 11 -24 576 35 0 0 49 14 196 35 0 0 35 0 0 35 0 0 15 -20 400 35 0 0 27 -8 64 ∑(x-ẋ)2 0 ∑(x-ẋ)2 2680
• 26. How to Calculate the Variance for Grouped Data 1. Calculate the mean. 2. Get the deviations by finding the difference of each midpoint from the mean. 3. Square the deviations and find its summation. 4. Substitute in the formula.
• 27. Find the Variance Class F Midpoint FMp _ _ _ _ Limits (2) (3) (4) X Mp - X (Mp-X)2 f( Mp-X)2 (1) 28-29 4 28.5 114.0 20.14 8.36 69.89 279.56 26-27 9 26.5 238.5 20.14 6.36 40.45 364.05 24-25 12 24.5 294.0 20.14 4.36 19.01 228.12 22-23 10 22.5 225.0 20.14 2.36 5.57 55.70 20-21 17 20.5 348.5 20.14 0.36 0.13 2.21 18-19 20 18.5 370.0 20.14 -1.64 2.69 53.80 16-17 14 16.5 231.0 20.14 -3.64 13.25 185.50 14-15 9 14.5 130.5 20.14 -5.64 31.81 286.29 12-13 5 12.5 62.5 20.14 -7.64 58.37 291.85 N= ∑fMp= ∑(Mp-X)2= 100 2,014.0 1,747.08
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