Concepts: step deviations, transformations and combined data

Statistics24 min readLesson 4 of 6

4.1 Step deviations Large mid-points make the variance calculation slow. The step-deviation method replaces each value by a small number. Choose an assumed mean a, usually a value near the middle of the data, and a common factor h, usually the class width. Define the step deviation d = (x - a)/h for each value. The true mean is then a plus h times the weighted average of the d values. The variance is h squared times the variance of the d values. The work is much lighter because the d values are small integers, and the same method works for the mean deviation once the true mean has been found.

d_{i} = \frac{x _{i} - a}{h}, x ˉ = a + h \frac{1}{N} \sum f_{i} d_{i}, σ^{2} = h^{2} [\frac{1}{N} \sum f_{i} d_{i}^{2} - (\frac{1}{N} \sum f_{i} d_{i})^{2}]

Step-deviation formulas for the mean and the variance

4.2 Worked example: step deviations Take mid-points 65, 75, 85, 95 and 105 with frequencies 2, 5, 8, 4 and 1. Choose a = 85 and h = 10. The d values are minus 2, minus 1, 0, 1 and 2. The weighted sum of the d values is minus 4 minus 5 plus 0 plus 4 plus 2, which is minus 3, and N is 20, so the weighted average is minus 0.15. The mean is 85 + 10 times minus 0.15 = 83.5. The weighted sum of the squared d values is 8 + 5 + 0 + 4 + 4 = 21, so the variance of the d values is 21/20 minus 0.0225 = 1.0275, and the variance of the data is 100 times that, about 102.75. The standard deviation is about 10.14. The same answer follows from the long method, but with much less arithmetic.

4.3 Effect of shifting and scaling Two transformation rules cover many exam questions. If every observation is increased by the same constant a, the mean increases by a while each deviation from the new mean is unchanged, so the variance and the standard deviation do not change. If every observation is multiplied by a constant k, both the mean and each deviation are multiplied by k, so the variance is multiplied by k squared and the standard deviation by the absolute value of k. These rules explain why a change of units changes the variance in a predictable way: converting marks to percentages by multiplying by two multiplies the variance by four.

y_{i} = x_{i} + a \Rightarrow σ_{y} = σ_{x}, y_{i} = k x_{i} \Rightarrow σ_{y} = ∣ k ∣ σ_{x}, σ_{y}^{2} = k^{2} σ_{x}^{2}

Effect of adding a constant and of multiplying by a constant on the standard deviation

4.4 Correcting a wrong mean or standard deviation Sometimes one observation is recorded wrongly and later corrected. The mean is easy to repair: add the difference between the correct and wrong values to the old total and divide by the number of observations. The standard deviation needs the sum of squares, which can be recovered from the old mean and standard deviation. If the wrong value w is replaced by the correct value c, the sum of observations changes by c - w and the sum of squares changes by c squared - w squared. Recomputing the mean and the variance from these corrected sums gives the right answer without redoing the whole table. The same technique handles the case where one observation is omitted or one extra observation is added.

4.5 Combining two data sets When two groups are merged, the combined mean is the weighted average of the two means, with the group sizes as weights. The combined variance is not simply the average of the two variances, because the two group means usually differ. The correct formula adds the average of the within-group variances to a term that depends on the squared distance between the group means. The second term is the variance of the group means about the combined mean. When the group means are equal the second term vanishes and the combined variance is the weighted average of the variances. This formula is a common source of exam questions on combined data.

x ˉ = \frac{n _{1} x ˉ _{1} + n _{2} x ˉ _{2}}{n _{1} + n _{2}}, σ^{2} = \frac{n _{1} σ _{1}^{2} + n _{2} σ _{2}^{2}}{n _{1} + n _{2}} + \frac{n _{1} n _{2} ( x ˉ _{1} - x ˉ _{2} ) ^{2}}{( n _{1} + n _{2} ) ^{2}}

Combined mean and variance when two data sets are merged

4.6 Worked example: combining two sets Take two groups: 4 observations with mean 10 and variance 4, and 6 observations with mean 20 and variance 9. The combined mean is (40 + 120)/10 = 16. The average within-group variance is (4 times 4 + 6 times 9)/10 = 70/10 = 7. The between-group term is (4 times 6 times (10 - 20) squared)/100 = 2400/100 = 24. The combined variance is 7 + 24 = 31 and the combined standard deviation is about 5.57. The between-group term is large because the two means are far apart, which is the point of the formula.

4.7 Exercise practice: NCERT questions worked in short 13.1 Question 1 For 4, 7, 8, 9, 10, 12, 13 and 17, the sum is 90 and the mean is 11.25. The absolute deviations are 7.25, 4.25, 3.25, 2.25, 1.25, 0.75, 1.75 and 5.75, whose sum is 26.5, so the mean deviation about the mean is 3.31. 13.2 Question 1 For 6, 7, 10, 12, 13, 4, 8 and 12, the sum is 72 and the mean is 9. The squared deviations are 9, 4, 1, 9, 16, 25, 1 and 9, whose sum is 74, so the variance is 74/8 = 9.25 and the standard deviation is about 3.04. 13.2 Question 2 For the first n natural numbers, the mean is (n + 1)/2 and the variance is (n squared - 1)/12. The remaining exercises 13.1 and 13.2 apply the same methods to frequency tables and continuous distributions.

Exam

In a combined-variance problem, compute the combined mean first and then use it in the second term. Leaving out the between-group term is the most common mistake and usually changes the answer a lot.

in hinglish

Bade numbers ke liye step deviation method use karo: d = (x - a)/h, phir mean a + h times average d, aur variance h squared times variance of d. Constant add karne se mean badalta hai lekin variance nahi; k se multiply karne se variance k squared guna hota hai. Agar koi observation galat likhi ho to sums correct karke naya mean aur variance nikaalo. Do groups combine karte waqt mean weighted average hota hai, aur variance me between-group term bhi add hota hai. Sirf dono variances ka average lena galat hai. Formula me group sizes zaroor daalo. Combined variance me between-group term ko ignore mat karo; agar dono groups ke means alag hain to woh term bada ho sakta hai aur answer galti se badal jaata hai. Variance aur SD ke answers ko approximate values se check karo, isse silly mistakes pakdi jaati hain.