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Descriptive Statistics – Central Tendency & Variability Chapter 3 (Part 2) MSIS 111 Prof. Nick Dedeke

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Learning Objectives Distinguish between measures of central tendency, measures of variability, measures of shape, and measures of association. Compute variance, standard deviation, and mean absolute deviation on ungrouped data. Differentiate between sample and population variance and standard deviation.

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Learning Objectives -- Continued Understand the meaning of standard deviation as it is applied by using the empirical rule and Chebyshev’s theorem. Compute the mean, mode, standard deviation, and variance on grouped data. Understand skewness, kurtosis, and box and whisker plots.

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Measures of Central Tendency: Ungrouped Data Measures of central tendency yield information about the center, or middle part, of a group of numbers. Common Measures of central tendency Mode Median Mean Percentiles Quartiles

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Mode The most frequently occurring value in a data set Applicable to all levels of data measurement (nominal, ordinal, interval, and ratio) Bimodal -- Data sets that have two modes Multimodal -- Data sets that contain more than two modes

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The mode is 44. 44 is the most frequently occurring data value. 35 37 39 40 41 43 44 45 46 48 Mode -- Example

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Median Middle value in an ordered array of numbers Applicable for ordinal, interval, and ratio data Not applicable for nominal data Unaffected by extremely large and extremely small values

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Median: Computational Procedure First Procedure Arrange the observations in an ordered array. If there is an odd number of terms, the median is the middle term of the ordered array. If there is an even number of terms, the median is the average of the middle two terms. Second Procedure The median’s position in an ordered array is given by (n+1)/2.

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Median: Example with an Odd Number of Terms Ordered Array 3 4 5 7 8 9 11 14 15 16 16 17 19 19 20 21 22 There are 17 terms in the ordered array. Position of median = (n+1)/2 = (17+1)/2 = 9 The median is the 9th term, which is 15. If the 22 is replaced by 100, the median is 15. If the 3 is replaced by -103, the median is 15.

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Median: Example with an Even Number of Terms Ordered Array 3 4 5 7 8 9 11 14 15 16 16 17 19 19 20 21 There are 16 terms in the ordered array. Position of median = (n+1)/2 = (16+1)/2 = 8.5 The median is between the 8th and 9th terms, 14.5. NOTE If the 21 is replaced by 100, the median is 14.5. If the 3 is replaced by -88, the median is 14.5.

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Arithmetic Mean Commonly called ‘the mean’ Is the average of a group of numbers Applicable for interval and ratio data Not applicable for nominal or ordinal data Affected by each value in the data set, including extreme values Computed by summing all values in the data set and dividing the sum by the number of values in the data set

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Population Mean Data for total population: 57, 57, 86, 86, 42, 42, 43, 56, 57, 42, 42, 43

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Mean for a Sample of 3

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Example: Computing Central Tend. Measures using Frequency Tables Mean= F i *X i F i = 1655/15 =110.33 XiXi FiFi F i * X i 552110 601 1003300 1255625 1404560 151655 Mode= 125 Median position = = (15+1)/2 = 8th Median value = 125

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Exercise: Computing Central Tend. Measures using Frequency Tables Mean= F i *X i F i= XiXi FiFi F i * X i 12 103 44 63 122 n=14 Mode= Median position = = Median value =

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Class intervalFrequency (F i ) Midpoints (M i ) [1 – 3) inches162 [3 – 5) inches24 [5 – 7) inches46 [7 – 9) inches38 [9 – 11) inches910 [11 – 13) inches612 40 Exercise: Central Tendency Measures for Grouped Data Modal class: Median position: Median class:

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Class intervalFrequency (F i ) Midpoint (M i ) (F i )*(M i ) [1 – 3) inches16232 [3 – 5) inches248 [5 – 7) inches4624 [7 – 9) inches3824 [9 – 11) inches91090 [11 – 13) inches61272 40 226 Example: Central Tendency Measures for Grouped Data Find the mean for the distribution: Mean: = (Σ F i *M i )/n = 226/40 = 5.65 inches

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Class intervalFrequency (F i ) Midpoint (M i ) (F i )*(M i ) [1 – 2) inches2 [2 – 3) inches2 [3 – 4) inches4 [4 – 5) inches2 [5 – 6) inches1 Exercise: Central Tendency Measures for Grouped Data Find the mean for the distribution: Mean: = (Σ F i *M i )/n = inches

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Exercise: Computing Central Tend. Measures using Frequency Tables XiXi FiFi F i * X i 122 448 6318 10330 12224 n=1482 XiXi FiFi F i * X i 020 100 4312 6530 10440 n=1482 Supplier 1 Supplier 2 We want to choose one of the two suppliers. We have data about their lateness in delivery (data is in hours). Which one has better statistical measures of central tendency?

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Measures of Dispersion: Variability Mean No Variability in Cash Flow (same amounts) Variability in Cash Flow (different amounts) Mean

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Measures of Variability: Ungrouped Data Measures of variability describe the spread or the dispersion of a set of data. Common Measures of Variability Range Interquartile Range Mean Absolute Deviation Variance Standard Deviation Z scores Coefficient of Variation

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Range The difference between the largest and the smallest values in a set of data Simple to compute Ignores all data points except the two extremes Example: Range = Largest - Smallest = 48 - 35 = 13 35 37 39 40 41 43 44 45 46 48

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Interquartile Range Range of values between the first and third quartiles Range of the middle 50% of the ordered data set Less influenced by extremes

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Deviation from the Mean Data set: 5, 9, 16, 17, 18 Mean: = 13 Deviations (X i - ) from the mean: -8, -4, 3, 4, 5 -8 -4 +3 +4 +5

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Mean Absolute Deviation Average of the absolute deviations from the mean 5 9 16 17 18 -8 -4 +3 +4 +5 0 +8 +4 +3 +4 +5 24 MAD X N.... 5 48 X

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Population Variance Average of the squared deviations from the arithmetic mean 2 2 130 5 260 X N. X 5 9 16 17 18 -8 -4 +3 +4 +5 0 64 16 9 16 25 130 X

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Population Standard Deviation Square root of the variance

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Sample Variance Average of the squared deviations from the arithmetic mean 2,398 1,844 1,539 1,311 7,092 625 71 -234 -462 0 390,625 5,041 54,756 213,444 663,866

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Sample Standard Deviation Square root of the sample variance

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Uses of Standard Deviation Indicator of financial risk Quality Control construction of quality control charts process capability studies Comparing populations household incomes in two cities employee absenteeism at two plants

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Exercise: Computing Standard Deviation using Frequency Tables XiXi FiFi F i * X i 020 140 4312 6330 10240 n=1482 Supplier 2 (mean = 5.8hours) Which one has better statistical measures of central tendency?

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Exercise: Computing Standard Deviation using Frequency Tables Mode= 4 hours Median position= 15/2 = 7.5 Median value= 6 hours Mean = 82/14 = 5.8 hours XiXi FiFi F i * X i 122 448 6318 10330 12224 n=1482 Supplier 1 (mean=5.8 hrs) Which one has better statistical measures of central tendency? Which supplier is better? Why?

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Standard Deviation as an Indicator of Financial Risk Annualized Rate of Return Financial Security A 15% 3% B 15%7%

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Variance and Standard Deviation of Grouped Data PopulationSample

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Population Variance and Standard Deviation of Grouped Data 1944 1152 44 1584 1452 1024 7200 20-under 30 30-under 40 40-under 50 50-under 60 60-under 70 70-under 80 6 18 11 3 1 50 25 35 45 55 65 75 150 630 495 605 195 75 2150 -18 -8 2 12 22 32 324 64 4 144 484 1024

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Measures of Shape Skewness Absence of symmetry Extreme values in one side of a distribution Kurtosis Peakedness of a distribution Leptokurtic: high and thin Mesokurtic: normal shape Platykurtic: flat and spread out Box and Whisker Plots Graphic display of a distribution Reveals skewness

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Relationship of Mean, Median and Mode

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Empirical Rule Data are normally distributed (or approximately normal) 95 99.7 68 Distance from the Mean Percentage of Values Falling Within Distance

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Chebyshev’s Theorem Applies to all distributions

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Chebyshev’s Theorem Applies to all distributions 1-1/3 2 = 0.89 1-1/2 2 =0.75 Distance from the Mean Minimum Proportion of Values Falling Within Distance Number of Standard Deviations K = 2 K = 3 K = 4 1-1/4 2 = 0.94

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Box and Whisker Plot Five specific values are used: Median, Q 2 First quartile, Q 1 Third quartile, Q 3 Minimum value in the data set Maximum value in the data set Inner Fences IQR = Q 3 - Q 1 Lower inner fence = Q 1 - 1.5 IQR Upper inner fence = Q 3 + 1.5 IQR Outer Fences Lower outer fence = Q 1 - 3.0 IQR Upper outer fence = Q 3 + 3.0 IQR

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Box and Whisker Plot Q1Q1 Q3Q3 Q2Q2 MinimumMaximum

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Exercises

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