Section 2.1-2.2 - Chapter 2: Sections 2.1-2.2 Numerical...

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Chapter 2: Sections 2.1-2.2 Numerical Summary of Measures In this lecture, we discuss (a) Some of the important characteristics or salient features of the data and for the population. (b) How to decide if a data is from a normal or a Weibull or a Poisson distribution? 2.1 Measures of Center First we discuss for data and then for the population distribution. 2.1.1 Means of Centre for Data 1. Sample Mean ( x ) The sample of mean of n observation x 1 , … x n is x = 1/n ∑ x i = ( x 1 + … + x n ) / n, where n denotes the number of observations. Example 1a . Suppose scores of 8 students in a test are : 35, 20, 45, 50, 42, 38, 39, 11. 1
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Then, the sample Mean = 280/8 = 35 (35%) Example 1b . Suppose, the last score is recorded, by mistake, as 71. Then, x = (269+71) / 8 = 340 / 8 = 42.5% About 22% increase in the sample mean. Note this is significant one. Rule: Increase one decimal place more than the one present in the data. In the above example, the data are in integers (no decimal places) and we denoted x = 42.5 (one place) 2. Median : % x This measure is less affected by outliers or extreme values. This divides the sample distribution in to two equal parts. Definition: Sample Median First order the observations from the smallest to the largest one . Then the median is defined as 2
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1 2 n + - th value, if n is odd x ~ = average of 2 n th and 1 2 n + th value, if n is even middle value, if n is odd = average of middle 2 value, if n is even Example 2 : The median of the values in Example 1a is: 11, 20, 35, 38, 39, 42, 45, 50 Here, n = 8 even; n/2 = 4. Take the middle values: 4 th and 5 th values. Hence, the median is x ~ = average of middle two values = {(38+39)/2} = 38.5 Example 3 : Find the median of Example 1b (one outlier case) Here, 20, 35, 38 , 39, 42, 45, 50, 71 3
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x ~ = (39+42) / 2 = 81/2 = 40.5. Remarks. (i) The median value is less affected than the mean. (ii) Also, this is an extreme case, as we replaced the smallest observation by one which is greater than the largest. (iii) Decreasing the first three smallest values or increasing the last three largest values in Example 3, does not affect the median. 2.Trimmed Mean (i) First order the observations (ordered data) from the Smallest to the largest. (ii) Let r n (0, 0.5). Then 100r% trimmed data is obtained by discarding the largest 100r% and the smallest 100r% of the data. 4
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Section 2.1-2.2 - Chapter 2: Sections 2.1-2.2 Numerical...

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