Numerical Measures of Central Tendency

Transcription

1

2 ҧ Numerical Measures of Central Tendency The central tendency of the set of measurements that is, the tendency of the data to cluster, or center, about certain numerical values; usually the Mean, Median or Mode. Mean: ҧ 1. Most common measure of central tendency 2. Acts as balance point 3. Affected by extreme values ( outliers ) 4. Denoted by x Formula: n x = σ i=1 n xi = x 1+ +x n n Where n= No. of observations & the i th observation is denoted by x i Example: Mean pulse rate = ( )/30=

3 Median: 1. The median is a measure of central tendency but it is a positional value. 2. When the data is ordered in ascending order, the median is the mid point of the data set. 3. n+1 The position of the Median is found at. Here however two situations arise. a. When n is odd: Ex: n=9, then position of Median is at 5= (9+1)/2 b. When n is even: Ex: n=10, then position of Median is at 5.5 = (10+1)/2 i.e. The Median is the average of the 5 th and 6 th value in the ordered data set. 4. The median is not affected by extreme values. Example: Raw Data: Ordered: Position of Median: Median=22.6 since n=5, (n+1)/2=3 and n is odd. Raw Data: Ordered: Position of Median: Median=( )/2=8.3 since n=6, (n+1)/2=3.5 and n is even. 2

4 Quartiles: Just like the median above, where we find the position by splitting the data into 2 parts, for quartiles we split the data into four parts. First Quartile (Q1): Median of the first half of the data Third Quartile (Q3): Median of the second half of the data Second Quartile (Q2) is the same as the median. Mode: 1. The mode is the value with the highest observed frequency. 2. It is not affected by extreme values. 3. In some cases data may have multiple modes. A mode need not necessarily be unique. Example: Raw Data: The value with the highest frequency, i.e. the value that appears most times in the data is 68. This can be verified from the dot plot on a previous slide. Uses of Mode: We can use the mode in various entrepreneurial scenarios where sales is considered. The most worn shoe size (because most people have mid-sized feet, it would be redundant to produce shoes that are too large or too small.

5 Effect of Linear Transformation Suppose every observation is multiplied by a fixed constant. Then median of transformed observations is the median of the original observations times that same constant. mean of transformed observations is the mean of the original observations times that same constant. Data: 10, 13, 18, 22, 29 Mean = Median = 18. Suppose transformed data = (-3)*original data. So transformed data: -30, -39, -54, -66, -87 Mean = (-3)*18.40 = Median = (-3)*18 =

6 Effect of Linear Transformation Suppose a fixed constant is added to (or subtracted from) each observation. Then median of transformed observations is the median of the original observations plus (or minus) that same constant. mean of transformed observations is the mean of the original observations plus (or minus) that same constant. Data: 10, 13, 18, 22, 29 Mean = Median = 18. Suppose transformed data = original data Hence transformed data: 12.5, 15.5, 20.5, 24.5, 31.5 Mean = = Median = =

7 Spread of a Distribution Are the values concentrated around the center of the distribution or they are spread out? Range, Interquartile Range, Variance, Standard Deviation. Note: Variance and standard deviation are more appropriate when the distribution is symmetric. 7 7

8 Range Range of the data is defined as the difference between the maximum and the minimum values. Data: 23, 21, 67, 44, 51, 12, 35. Range = maximum minimum = = 55. Disadvantage: A single extreme value can make it very large, giving a value that does not really represent the data overall. On the other hand, it is not affected at all if some observation changes in the middle. 8 8

9 Interquartile Range (IQR) What is IQR? IQR = Third Quartile (Q 3 ) First Quartile (Q 1 ). What are quartiles? Recall: Median divides the data into 2 equal halves. The first quartile, median and the third quartile divide the data into 4 roughly equal parts. 9 9

10 Quartiles The first quartile (Q 1, lower quartile) is that value which is larger than 25% of observations, but smaller than 75% of observations. The second quartile (Q 2 ) is the median, which is larger than 50% of observations, but smaller than 50% of observations. The third quartile (Q 3, upper quartile) is that value which is larger than 75% of observations, but smaller than 25% of observations. Obviously, Q 1 < Q 2 (= median) < Q 3. How to compute the quartiles? We shall use TI 83/84 Plus

11 TI 83/84 Plus commands To enter the data: Press [STAT] Under EDIT select 1: Edit and press ENTER Columns with names L1, L2 etc. will appear Type the data value under the column; each data entry will be followed by ENTER. To clear data: Pressing CLEAR will clear the particular data. To clear all data from all columns press [2nd] & + and then choose 4: ClrAllLists

12 TI 83/84 Plus commands 12 12

13 IQR vs. Range IQR is a better summary of the spread of a distribution than the range because it has some information about the entire data, where as range only has information on the extreme values of the data. IQR is less outlier-sensitive than range

14 Outlier-sensitivity Data: 10, 13, 17, 21, 28, 32 Without the outlier IQR = 15 Range = 22 Data: 10, 13, 17, 21, 28, 32, 59 With the outlier IQR = 19 Range = 49 Conclusion: IQR is less outlier-sensitive than range

15 Variance and Standard Deviation The sample variance (s 2 ) is defined as: s ( x1 x) ( xn x). n 1 Subtract the mean from each value, square each difference, add up the squares, divide by one fewer than the sample size. The sample standard deviation (s), is the positive square root of sample variance, i.e. s 2 s

16 Variance and Standard Deviation Larger the variance (and standard deviation) more dispersed are the observations around the mean. The unit of variance is square of the unit of the original data, whereas standard deviation has the same unit as the original data. Both variance and standard deviation are more appropriate for symmetric distributions

17 Standard Deviation: An Example Data: 3, 12, 8, 9, 3 (n=5 in this case) Mean = ( )/5 = 35/5 =7. Data Deviations from mean Squared Deviations = -4 (-4)x(-4) = = 5 5 x 5 = = 1 1 x 1 = = 2 2 x 2 = = -4 (-4)x(-4) = Total = 62 Now divide by n-1=4: s 2 = 62/4 = s = 15.5 = Answer: The standard deviation in this example is 3.94 and the variance is

18 Effect of Linear Transformation Suppose every observation is multiplied by a fixed constant. Then range/iqr/standard deviation of transformed observations is the range/iqr/standard deviation of the original observations times the absolute value of that same constant. variance of transformed observations is the variance of the original observations times the square of that same constant. Temperature data (in F): 10, 13, 18, 22, 29 Range = 19 F, IQR =14 F, s = 7.5 F, s 2 = F 2. Suppose transformed data = (-3)*original data. So transformed data (in F): -30, -39, -54, -66, -87 Range = -3 *19 = 57 F, IQR = -3 *14 = 42 F, s = -3 * 7.5 = F, s 2 = (-3) 2 *56.25 = F

19 Effect of Linear Transformation Suppose a fixed constant is added to (or subtracted from) each observation. Then range/iqr/standard deviation/variance of transformed observations remains the same as that of the original observations. Temperature data (in F): 10, 13, 18, 22, 29 Range = 19 F, IQR =14 F, s = 7.5 F, s 2 = F 2. Suppose transformed data = original data Hence transformed data (in F): 12.5, 15.5, 20.5, 24.5, 31.5 Range = 19 F, IQR =14 F, s = 7.5 F, s 2 = F

20 Chebyshev s rule For any distribution at least 1 1 of the k observations will fall within k standard 2 deviations of mean, where k 1. Chebyshev s rule is for any distribution, whereas the empirical rule is valid only for approximately symmetric unimodal (mound-shaped) distribution. If k=1, not much information is available from Chebyshev s rule. According to Chebyshev at least 75% observations fall within 2 standard deviations of mean. According to Chebyshev at least 88.9% of observations fall within 3 standard deviations of mean. 20

21 Empirical rule For approximately symmetric unimodal (bellshaped/mound shaped) distribution Approximately 68% of observations fall within 1 standard deviation of mean. Approximately 95% of observations fall within 2 standard deviations of mean. Approximately 99.7% of observations fall within 3 standard deviations of mean. 21

22 Empirical rule 22

23 Empirical rule 23

24 Chebyshev s Rule can be applied for any distribution. Performs bad in unimodal symmetric distribution. Empirical rule can be applied only to unimodal symmetric distribution. Performs good in this shape of distribution.