Descriptive Statistics

Transcription

1 Sherif Khalifa Sherif Khalifa () Descriptive Statistics 1 / 34

2 Definition Measures of central tendency yield information about the center, or middle part, of a group of numbers. Mode Median Mean Percentiles Quartiles Sherif Khalifa () Descriptive Statistics 2 / 34

3 Definition The mode is the most frequently occuring value in a set of data. When there are two modes, the data are said to be bimodal. Datasets with more than two modes are referred to as multimodal The mode = 19 Sherif Khalifa () Descriptive Statistics 3 / 34

4 Definition The median is the middle value in an ordered array of numbers. Definition The median is the center number, or with an even number of observations, the average of the middle two terms. Definition For an array with an odd number of terms, the median is the middle number. Definition For an array with an even number of terms, the median is the average of the two middle numbers. Sherif Khalifa () Descriptive Statistics 4 / 34

5 The median = The median = = 14.5 Sherif Khalifa () Descriptive Statistics 5 / 34

6 Definition The mean is the average of a group of numbers. µ = x i N = = 18.6 Sherif Khalifa () Descriptive Statistics 6 / 34

7 Shopping Centre Size (1000 m 2 ) MetroCentre Trafford Centre Westfield Stratford City Bluewater Liverpool One Westfield London Intu Merry Hill Manchester Arndale Meadowhall Lakeside St. David s Bullring Eldon Square Sherif Khalifa () Descriptive Statistics 7 / 34

8 The mode = The median = The mean = = Sherif Khalifa () Descriptive Statistics 8 / 34

9 Definition Percentiles are measures of central tendency that divide a group of data into 100 parts. Organize the numbers into an ascending order array. Calculate the percentile location (i) by:i = P 100 (N) If i is a whole number, the Pth percentile is the average of the value at the ith location and the value at the (i + 1)st location. If not, the Pth percentile value is located at the whole number part of (i + 1). Sherif Khalifa () Descriptive Statistics 9 / 34

10 i = 30 (8) = i = 3 P 30 = 13 i = (8) = P 50 = = Sherif Khalifa () Descriptive Statistics 10 / 34

11 Definition Quartiles are measures of central tendency that divide a group of data into four subgroups. Sherif Khalifa () Descriptive Statistics 11 / 34

12 i = (8) = 2 Q 1 = P 25 = i = (8) = 4 Q 2 = P 50 = i = (8) = 6 Q 3 = P 75 = ( ) 2 ( ) 2 ( ) 2 = = = Sherif Khalifa () Descriptive Statistics 12 / 34

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15 Definition Measures of variability describe the spread or the dispersion of a set of data. Range Interquartile Range Mean Absolute Deviation Variance Standard Deviation Z-scores Coeffi cient of variation Sherif Khalifa () Descriptive Statistics 15 / 34

16 Distributions may have the same mean but different variability or dispersions. Sherif Khalifa () Descriptive Statistics 16 / 34

17 Definition The range is the difference between the largest value of a data set and the smallest value of a set. Range = = Sherif Khalifa () Descriptive Statistics 17 / 34

18 Definition The interquartile range is the range of values between the first and third quartile. Country Exports ($ billion) Canada Mexico China Japan 66.8 United Kingdom 53.8 Germany 49.4 South Korea 44.5 Netherlands 43.1 Brazil 42.4 Hong Kong 40.9 Belgium 34.8 France 31.3 Singapore 30.2 Taiwan 26.7 Switzerland 22.2 Sherif Khalifa () Descriptive Statistics 18 / 34

19 i = 25 (15) = Q 1 = P 25 = 30.6 i = 75 (15) = Q 3 = P 75 = 70.0 Q 3 Q 1 = = 39.4 Sherif Khalifa () Descriptive Statistics 19 / 34

20 Definition Mean absolute deviation is the average of the absolute values of the deviations around the mean of a set of numbers. x x µ x µ MAD = x i µ n = 24 5 = 4.8 Sherif Khalifa () Descriptive Statistics 20 / 34

21 Definition The variance is the average of the squared deviations about the arithmetic mean for a set of numbers. Definition The standard deviation is the square root of the variance. Sherif Khalifa () Descriptive Statistics 21 / 34

22 x x µ (x µ) σ 2 = (x i µ) 2 = 130 N 5 = 26 (x i µ) 2 σ = = 5.1 N Sherif Khalifa () Descriptive Statistics 22 / 34

23 Firm x (x x) 2 Deloitte & Touche Ernst Young Price Waterhouse Coopers KPMG RSM Grant Thornton x = x i n = 9524 = s 2 = (x i x) 2 n 1 = σ = = = Sherif Khalifa () Descriptive Statistics 23 / 34

24 Definition A Z score represents the number of standard deviations a value is above or below the mean of a set of numbers when the data are normally distributed. z = x i µ σ x i = 70 µ = 50 σ = 10 z = = 2 Sherif Khalifa () Descriptive Statistics 24 / 34

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27 Definition The coeffi cient of variation is the ratio of the standard deviation to the mean expressed in percentage. CV = σ µ (100) Sherif Khalifa () Descriptive Statistics 27 / 34

28 Stock A Stock B µ A = 64.4; µ B = 13 σ A = 4.84; σ B = 3.03 CV A = σ (100) = 7.5% µ CV B = σ (100) = 23.3% µ Sherif Khalifa () Descriptive Statistics 28 / 34

29 µ A = 29; µ B = 84 σ A = 4.6; σ B = 10 CV A = σ (100) = 15.85% µ CV B = σ (100) = 11.90% µ Sherif Khalifa () Descriptive Statistics 29 / 34

30 Grouped Data Grouped data do not provide information about individual values. Measures of central tendency and variability for grouped data must be computed differently from those of raw data. Sherif Khalifa () Descriptive Statistics 30 / 34

31 Grouped Data f i M i f i M i Cf (M i µ) (M i µ) 2 f i (M i µ) 2 20-under under under under under under Sherif Khalifa () Descriptive Statistics 31 / 34

32 Grouped Data µ = f i M i = 2150 f i 50 = 43 ( N ) 2 cfp Median = L + (W ) = 40 + f med Mode = = 35 2 σ 2 = f i (M i µ) 2 f i = = 144 σ = 144 = 12 ( 50 ) 2 24 (10) = Sherif Khalifa () Descriptive Statistics 32 / 34

33 Grouped Data f i M i f i M i (M i µ) (M i µ) 2 f i (M i µ) 2 1-under under under under under under Sherif Khalifa () Descriptive Statistics 33 / 34

34 Grouped Data µ = f i M i = 416 f i 60 = 6.93 ( N ) ( 2 60 ) cfp 2 29 Median = L + (W ) = 7 + (2) = f med Mode = = 8 2 σ 2 = f i (M i µ) 2 = = f i 60 σ = Sherif Khalifa () Descriptive Statistics 34 / 34