Data Summary Using Descriptive Measures Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Types of Descriptive Measures Central Tendency Variation Position Shape Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Measures of Central Tendency Mean Median Midrange Mode Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

The Mean The Mean is simply the average of the data. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Sample Mean Each value in the sample is represented by x thus to get the mean simply add all the values in the sample and divide by the number of values in the sample Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Accident Data Set Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

The Median The Median (Md) of a set of data is the value in the center of the data values when they are arranged from lowest to highest. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Accident Data Ordered array: 5, 6, 7, 9, 23 The value that has an equal number of items to the right and left is the median. Thus Md = 7 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

The Median In general if n is odd, Md is the center data value of the ordered data set. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Accident Data Ordered array: 5, 6, 7, 9, 23 Md  5  1 2       st ordered value = 3rd value Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

The Median If n is even, Md is the average of the two center values of the ordered data set. For the ordered data set: 3, 8, 12, 14 Md  8  12 2       = 10.0 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

The Midrange The Midrange (Mr) provides an easy-to- grasp measure of central tendency. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Accident Data Mr  5  23 2 = 14.0 x  Md = 7 Note: that the Midrange is severely affected by outliers Compare: Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

The Mode The Mode (Mo) of a data set is the value that occurs more than once and the most often. The Mode is not always a measure of central tendency; this value need not occur in the center of the data. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Level of Measurement and Measure of Central Tendency Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Measures of Variation Homogeneity refers to the degree of similarity within a set of data. The more Homogeneous a set of data is, the better the mean will represent a typical value. Variation is the tendency of data values to scatter about the mean,. x Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Common Measures of Variation Range Variance Standard Deviation Coefficient of Variation Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

The Range For the Accident data: Range = H - L = = 18 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

The Variance and Standard Deviation Both measures describe the variation of the values about the mean. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Accident Data Data Value(x - ) (x - ) = 220 x x ( x – x ) 2  Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Definition: Sample Variance Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Definition: Sample Standard Deviation s  55.0  Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Definition: Population Variance Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Definition: Population Standard Deviation  ( x –  ) 2  N Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

The Coefficient of Variation The Coefficient of Variation (CV) is used to compare the variation of two or more data sets where the values of the data differ greatly. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Example Data Set 1: 5, 6, 7, 9, 23 Data Set 2: 5000, 6000, 7000, 9000, 23,000 CV  Data Set = CV  7, ,000. = Data Set 2 Thus both data sets exhibit the same relative variation Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Measures of Position Percentile (Quartile) Z Score Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Percentile The 35th Percentile (P 35 ) is that value such that at most 35% of the data values are less than P 35 and at most 65% of the data values are greater than P 35. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Percentile Texon Industries Data 17.5 represents the position of the 35th percentile Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Percentile: Location Rules If n  P/100 is not a counting number, round it up, and the Pth percentile will be the value in this position of the ordered data. If n  P/100 is a counting number, the Pth percentile is the average of the number in this location (of the ordered data) and the number in the next largest location. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Quartiles Quartiles are merely particular percentiles that divide the data into quarters, namely: Q 1 = 1st quartile = 25th percentile (P 25 ) Q 2 = 2nd quartile = 50th percentile (P 50 ) Q 3 = 3rd quartile = 75th percentile (P 75 ) Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Z Scores Z score determines the relative position of any particular data value x and is based on the mean and standard deviation of the data set. The Z score is expresses the number of standard deviations the value x is from the mean. A negative Z score implies that x is to the left of the mean and a positive Z score implies that x is to the right of the mean. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Z Score Equation Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Measures of Shape Skewness Kurtosis Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Skewness Skewness measures the tendency of a distribution to stretch out in a particular direction Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Skewness In a symmetrical distribution the mean, median, and mode would all be the same value. Sk = 0 (fig 3.7) A positive Sk number implies a shape which is skewed right (fig3.8). The mode < median < mean In a data set with a negative Sk value (fig3.9) the mean < Median < Mode Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Figure 3.7 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Figure 3.8 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Figure 3.9 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Skewness Calculation Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Kurtosis Kurtosis measures the peakedness of the distribution. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Chebyshev’s Inequality At least 75% of the data values are between x - 2s and x + 2s or At least 75% of the data values have a Z score value between -2 and +2 At least 89% of the data values are between x - 3s and x + 3s In general, at least (1-1/k 2 ) x 100% of the data values lie between x - ks and x + ks for any k>1 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Empirical Rule Under the assumption of a bell shaped population Approximately 68% of the data values lie between Approximately 95% of the data values lie between Approximately 99.7% of the data values lie between Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Chebyshev’s versus Empirical Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing

Grouped Data Approximations Where: f is the frequency of the class and m is the m is the midpoint of the class Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing