Measures of Variability.pptx

measure of variablity

1. Measures of Variability: (The range, Quartile Deviation, Average Deviation and standard deviation ) By: - Dr. Satish P. Pathak Department of Education (CASE) Faculty of Education and Psychology, The Maharaja Sayajirao University of Baroda, Vadodara (Gujarat) ………………………………………………………………………

2. Measures of Variability The terms variability, spread, and dispersion are synonyms, and refer to how spread out a distribution is. Mean = 7 in both the cases Scores are more densely packed Scores are more spread out

3. Measures of Variability How far the scores have shown spread out from the mean? Dispersion within a dataset can be measured or described in several ways by using Measures of Variability. It will make the distribution and interpretation more meaningful. It shows the specific nature of distribution of data.

4. Measures of Variability There are four major “ Measures of Variability ”: The Range The Quartile Deviation The Mean or Average Deviation The Standard Deviation

5. (1) : The Range The simplest measure of variability Range = The difference between the highest score and lowest score The range is useful for showing the spread within a dataset and for comparing the spread between similar datasets.

6. Selection and Application of Range The Range is used when ; the data are too scant (little) or too scattered only an idea of extreme scores or of total spread is wanted

7. Limitations It is very sensitive to the smallest and largest data values. It is not a stable statistics as its value can differ from sample to sample drawn from the same population. In order to reduce the problems caused by outliers in a dataset, the inter-quartile range is often calculated instead of the range. [IQR : It is the range for the middle 50% of the data. It overcomes the sensitivity to extreme data values. ]

8. Quantiles The extensions of the Median concept because they are values which divide a set of data into equal parts. Median : Divides the distribution into two equal parts. Quartile : Divides the distribution into four equal parts. Decile : Divides the distribution into ten equal parts. Percentile : Divides the distribution into one hundred equal parts.

9. (2) : The Quartile Deviation : Q Q₁ Q₂ Q₃ Inter-quartile Range Median 25 th Percentile 75 th Percentile Since IQR includes middle 50 % of scores, the value of Q gives clear picture of spread / dispersion. Q₁ : 1 st Quartile The point below Which 25 th per cent of the scores lie Q₃ : 3 rd Quartile The point below Which 75 th per cent of the scores lie

10. The Quartile Deviation : Q When the extreme scores in the given distribution are very high and very low, the range will be very high. The inter-quartile range provides a clearer picture of the overall dataset by removing/ignoring the outlying values. The Quartile deviation is one-half the scale distance between the 75 th and 25 th percentiles in a frequency distribution. (i.e. Semi- interquartile Range)

11. The Quartile Deviation : Q If the middle 50% of scores in the distribution are densely packed, quartiles will be nearer to each other & value of Q will be less. If the middle 50 % of scores in the distribution are more spread out, quartiles will be far from each other & value of Q will be high.

12. The Quartile Deviation : Q e.g. ( i ) 10,10,65,100,120, 180,200, 270,300,500 (n = 10) Upper half 180,200, 270 ,300,500 Q₃ = 270 Lower half 10,10, 65 ,100,120 Q₁ = 65 IQR = Q₃ − Q₁ = (270 − 65 ) = 205 Q = ( Q₃ − Q₁ ) / 2 = (270 − 65 ) / 2 = 205 / 2 = 102.5 Mathematically, Q = ( Q₃ − Q₁ ) / 2

13. The Quartile Deviation : Q (For ungrouped data) e.g. (ii) 22,25,34,35,41,41,46 , 46 , 46,47,49,54,54,59,60 (n = 15) Upper half (including Median) 46 , 46,47, 49,54 ,54,59,60 Q₃ = 49 + 54 / 2 = 51.5 Lower half (including Median) 22,25,34, 35,41 ,41,46 , 46 Q₁ = 35 + 41 / 2 = 38 IQR = Q₃ − Q₁ = 51.5 − 38 = 13.5 Q = ( Q₃ − Q₁ ) / 2 = (51.5 − 38 ) / 2 = 13.5 / 2 = 6.75

14. The Quartile Deviation : Q (For Grouped Data) Scores Exact Units of Class Interval f F 52 – 55 51.5 – 55.5 1 65 48 – 51 47.5 – 51.5 64 44 - 47 43.5 – 47.5 5 64 40 - 43 39.5 – 43.5 10 59 36 – 39 35.5 – 39.5 20 49 @ 32 - 35 31.5 – 35.5 12 29 28 - 31 27.5 – 31.5 8 17 # 24 – 27 23.5 – 27.5 2 9 20 – 23 19.5 – 23.5 3 7 16 - 19 15.5 – 19.5 4 4 N = 65 # : Which contains the Q₁ @ : Which contains the Q₃

15. The Quartile Deviation : Q (For Grouped Data) Q₁ = L + N /4 − F x i f Q₃ = L + 3N /4 − F x i f N / 4 = 65 / 4 = 16.25 3N / 4 = 3x65 / 4 = 48.75 Where, L = The exact lower limit of the interval in which the Quartile falls i = The length of the interval F = Cumulative frequency below the interval which contains the Quartile f = The frequency of the interval containing the Quartile N = Total number of observations

16. The Quartile Deviation : Q (For Grouped Data) Q₁ = L + N /4 − F x i f = 27.5 + 16.25 − 9 x 4 8 = 27.5 + 3.625 = 31.125 Q₃ = L + 3N /4 − F x i f = 35.5 + 48.75 − 29 x 4 20 = 35.5 + 3.95 = 39.45 Q = ( Q₃ − Q₁ ) / 2 = 39.45 − 31.125 = 4.16 2

17. Selection and Application of the Q The Quartile Deviation is used when ; only the median is given as the measure of central tendency; there are scattered or extreme scores which would influence the S.D. excessively; the concentration around the Median, the middle 50 % scores , is of primary interest.

18. A Deviation score A score expressed as its distance from the Mean is called a deviation score. x = ( X − ) e.g. 6, 5, 4, 3, 2, 1 Mean ( ) = 21/6 = 3.50 [ e.g. 6 – 3.50 = 2.5 is a deviation score of 6 ] Sum of deviations of each value from the mean : 2.5 + 1.5 + 0.5 + (- 0.5) + (- 1.5 ) + (- 2.5 ) = 0 i.e. ∑ ( X − ) = 0 ∑ x = 0 Definition of the Mean : The Mean is that value in a distribution around which the sum of the deviation score equals zero.

19. (3) : The Average Deviation : AD or Mean Deviation (MD) AD is the mean of the deviations of all observations taken from their mean. In averaging deviations, to find AD, the signs ( + and − ) are not taken into consideration i.e. all the deviations are treated as positive.

20. The Average Deviation : AD (For ungrouped data) X : Marks obtained x Deviation │ x │ 18 − 5 5 19 − 4 4 21 − 2 2 19 − 4 4 27 + 4 4 31 + 8 8 22 − 1 1 25 + 2 2 28 + 5 5 20 − 3 3 ∑ X = 230 ∑ x = 0 ∑ │x│ = 23 Mean = ∑ X / N = 230 / 10 = 23 Average Deviation = ∑ │x│ / N = 23 / 10 = 2.3

21. The Average Deviation : AD ( For grouped data ) : (Under Assumed Mean Method) Scores Class Interval Exact units of Class Interval Mid -Point x f x‘ Devi. fx ' 60-69 59.5 – 69.5 64.5 1 3 3 50-59 49.5 – 59.5 54.5 4 2 8 40-49 39.5 – 49.5 44.5 10 1 10 30-39 29.5 – 39.5 34.5 15 20-29 19.5 – 29.5 24.5 8 – 1 – 8 10-19 9.5 – 19.5 14.5 2 – 2 – 4 N = 40 ∑ │ fx ’│ = 33 Average Deviation = ∑ │ fx ’│ / N = 33 / 40 = 0.825

22. Selection and Application of the AD AD is used when : It is desired to consider all deviations from the mean according to their size; Extreme deviations would effect standard deviation excessively.

23. Limitations : A.D. It is based on all deviations, therefore it may be increased because of one or more extreme deviation/s. All the deviations are treated as positive. Needs long mathematical calculations. Hence, it is rarely used.

24. The Variance The sum of the squared deviations from the mean , divided by N , is known as the Variance . : OR This value describes characteristics of distribution. It will be employed in a number of very important statistical tests. This value is too large to represent the spread of scores because of squaring the deviations.

25. (4) : The Standard Deviation : σ The S.D. is the most general and stable measure of variability. The S.D. is the positive square root of the variance. The Standard Deviation is a measure of how spread out numbers are. The symbol for Standard Deviation is σ (the Greek letter sigma).

26. The Standard Deviation : Formulas The  Population  Standard Deviation:   The  Sample  Standard Deviation:    The important change is  "N-1" instead of "N"  (which is called "Bessel's correction” - Friedrich Bessel ). [ The factor  n /( n  − 1) is itself called Bessel's correction.]

27. Calculation of SD Example : Ram has 20 Rose plants. The number of flowers on each plant is 9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4, 10, 9, 6, 9, 4 Work out the Standard Deviation. ************ Step 1 . Work out the mean In the formula above  μ  (the Greek letter "mu") is the  mean

28. Calculation of SD Mean (µ) = ∑ X / N = 140 / 20 = 7 Step 2 . Then for each number: subtract the Mean and square the result This is the part of the formula that says: Example (continued): (9 - 7) 2  = (2) 2  = 4 (2 - 7) 2  = (-5) 2  = 25 (5 - 7) 2  = (-2) 2  = 4 ……… etc….

29. Calculation of SD Step 3 . Then work out the mean of those squared differences . = 4+25+4+9+25+0+1+16+4+16+0+9+25+4+9+9+4+1 +4+9 =  178 Mean of squared differences = (1/20) × 178 = 8.9 (Note: This value is called the "Variance")

30. Calculation of SD Step 4 . Take the square root of the Variance : Example (concluded): σ = √ (8.9) =  2.983... But, ... sometimes our data is only a  sample  of the whole population.

31. Calculation of SD (For the Sample) Example : Ram has 20 rose plants, but what if Ram only counted the flowers on 6 of them? The "population" is all 20 rose plants, and the "sample" is the 6 he counted. Let us say they are: 9, 2, 5, 4, 12, 7 = 6.5 s = √ (13.1) =  3.619...

32. Comparison Comparison of… N Mean Standard Deviation Population 20 7 2.983 Sample 06 6.5 3.619 Sample Mean is wrong by 7% Sample Standard Deviation is wrong by 21% When we take a sample, we lose some accuracy.  

33. Calculation of SD ( For ungrouped data ) Score (X) x or X − x ² 15 1 1 10 − 4 16 15 1 1 20 6 36 8 − 6 36 10 − 4 16 25 11 121 9 − 5 25 ∑ x ² = 252 Mean ( ) = ∑ X / N = 112 / 8 = 14 = 252 / 8 = √ 31.8 = 5.64

34. Exercise Calculate the Mean, Quartile deviation, Average deviation and Standard deviation for the given ungrouped data. 41, 47, 48, 50, 51, 53, 60 Reveal your answer. Compute S.D. for the given data: 18, 25, 21, 19, 27, 31, 22, 25, 28, 20

35. Calculation of SD ( Direct method without using deviation ) Raw Scores : x x ² 15 225 10 100 15 225 20 400 8 64 10 100 25 625 9 81 ∑ x = 112 ∑ x ² = 1820 σ = √ N ∑ x ² − ( ∑ x )² N = √ 8 x 1820 − (112) ² 8 = 5.612

36. Calculation of Mean and SD ( For grouped data : Based on Frequency Distribution) C.I. Midd . Pt. : X f x: Devi. fx fx ² 80-84 82 5 75-79 77 6 70-74 72 8 65-69 67 10 60-64 62 16 55-59 57 20 50-54 52 12 45-49 47 9 40-44 42 8 35-39 37 6 100 σ = √ N ∑ f x ² N x : Deviation of each Middle point from Mean Mean = ∑ f . X / N = 58.55 σ = 11.78

37. Calculation of Mean and SD ( For grouped data : Assumed Mean Method) C.I. Mid. Pt. : X f x ´ fx ´ fx ´² 52-55 53.5 1 4 4 16 48-51 49.5 3 44-47 45.5 5 2 10 20 40-43 41.5 10 1 10 10 36-39 37.5 A.M. 20 32-35 33.5 12 − 1 − 12 12 28-31 29.5 8 − 2 − 16 32 24-27 25.5 2 − 3 − 6 18 20-23 21.5 3 − 4 − 12 48 16-19 17.5 4 − 5 − 20 100 N = 65 σ = i √ N ∑ f x ´ ² − (∑ f x ´ ) ² N σ = 7.51 i = length of class interval …………. COMPLETE IT

38. Selection and Application of S.D. S.D. is used when : the statistics having greatest stability is required; extreme deviations exercise a proportionally greater effect upon the variability; co-efficient of correlation and other statistics are subsequently computed.

39. Thank You