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# Central tendency

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### Central tendency

1. 1. Measures of Central TendencyDR MUHAMMAD TAUSEEF JAVEDMBBS.DPH.FCPS(CM)M.Phil(CM).Dip-Card DR TAUSEEF JAVED
2. 2. StatisticsMeasures of Central Tendency DR TAUSEEF JAVED
3. 3. • Measures of central tendency are scores that represent the center of the distribution.• Three of the most common measures of central tendency are: – Mean – Median – Mode DR TAUSEEF JAVED
4. 4. Mean• The most commonly used measure of central tendency• When people ask about the “average” of a group of scores, they usually are referring to the mean.• The mean is the sum of all the scores in the distribution divided by the total scores (the mathematical average). DR TAUSEEF JAVED
5. 5. Mean (con’t)• Mean of a sample• Mean of a population DR TAUSEEF JAVED
6. 6. Mean (con’t) Exam Scores75 82 72 68 89 X sum all scores91 78 94 88 75 n = total number of scores for the sample DR TAUSEEF JAVED
7. 7. Mean (con’t) Performance and Memory S tudy2 2 3 3 4 6 5 Frequency4 4 4 4 1 4 3 0 2 1 0 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 Number of Words Recalled The mean includes the weight of every score. DR TAUSEEF JAVED
8. 8. Pros and cons of using mean• Pros – Summarizes data in a way that is easy to understand. – Uses all the data – Used in many statistical applications• Cons – Affected by extreme values• E.g., average salary at a company – 12,000; 12,000; 12,000; 12,000; 12,000; 12,000; 12,000; 12,000; 12,000; 12,000; 20,000; 390,000 – Mean = \$44,167 DR TAUSEEF JAVED
9. 9. Median• The middle score of the distribution when all the scores have been ranked.• If there are an even number of scores, the median is the average of the two middle scores. DR TAUSEEF JAVED
10. 10. Median (con’t)2 Number of Words Recalled in Performance Study 2 3 3 4 4 4 4 410 DR TAUSEEF JAVED
11. 11. Mode• The most frequent score in the distribution.• A distribution where a single score is most frequent has one mode and is called unimodal.• A distribution that consists of only one of each score has n modes.• When there are ties for the most frequent score, the distribution is bimodal if two scores tie or multimodal if more than two scores tie. DR TAUSEEF JAVED
12. 12. DR TAUSEEF JAVED
13. 13. Mode (con’t)2 2 Number of Words Recalled in Performance Study 3 3 The mode is 4. 4 4 4 4 410 DR TAUSEEF JAVED
14. 14. Mode (con’t) 72 72 73 76 78 81 83 85 85 86 87 88 90 91 92 This distribution is bimodal.Demonstration DR TAUSEEF JAVED
15. 15. Red Blue Green YellowYellow Green Blue RedGreen Red Yellow Blue DR TAUSEEF JAVED
16. 16. Red Blue Green YellowYellow Green Blue RedGreen Red Yellow Blue DR TAUSEEF JAVED
17. 17. • Calculations• Key: dependent measure is reaction time – Time it takes to say the color• Determine the mean, median, and mode of the datasets in the handout. DR TAUSEEF JAVED
18. 18. The best measure of central tendency depends on…. • The scale of measurement. • The shape of the distribution. DR TAUSEEF JAVED
19. 19. Scales of Measurement• Nominal scale = mode• Ordinal scale = median• Interval scale = mean, median, or mode• Ratio scale = mean, median, or mode DR TAUSEEF JAVED
20. 20. Shape of the DistributionSkew refers to the general shape of a distributionwhen it is graphed. Symmetrical = zero skew Scores clustered on the high or low end of a distribution = skewed distribution DR TAUSEEF JAVED
21. 21. Symmetrical Distribution 16 14 Frequency 12 10 8 6 4 2 0 24.5 29.5 34.5 39.5 44.5 49.5 54.5 59.5 64.5 69.5 ScoresThe mean, median, and mode are the same. DR TAUSEEF JAVED
22. 22. • The normal distribution is the “ideal” symmetrical distribution DR TAUSEEF JAVED
23. 23. Distributions that are skewed have oneside of the distribution where the data frequency tapers off DR TAUSEEF JAVED
24. 24. Skewed Distribution Positive Skew 12 10 Frequency 8 6 4 2 0 27 32 37 42 47 52 57 62 67 72 77 ScoresTail points in the positive direction. DR TAUSEEF JAVED
25. 25. Skewed Distribution Negative Skew 12 10Frequency 8 6 4 2 0 27 32 37 42 47 52 57 62 67 72 77 Scores Tail points in negative direction. DR TAUSEEF JAVED
26. 26. The mean will either underestimate or overestimate the center of skewed distributions. Positive Skew Negative Skew 12 12 10 10Frequency Frequency 8 8 6 6 4 4 2 2 0 27 32 37 42 47 52 57 62 67 72 77 0 27 32 37 42 47 52 57 62 67 72 77 Scores Scores Mode Mode Median Median Mean Mean DR TAUSEEF JAVED
27. 27. Kurtosis• Measure of the flatness or peakedness of the distribution DR TAUSEEF JAVED