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Central tendency

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Tauseef Jawaid
Tauseef Jawaid

This document discusses measures of central tendency including the mean, median, and mode. It provides examples and definitions for each measure. The mean is the average and is calculated by summing all values and dividing by the total number. The median is the middle value when values are ranked in order. The mode is the most frequent value. The best measure depends on the scale of measurement and shape of the distribution, such as whether it is symmetrical or skewed.

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Measures of Central Tendency DR MUHAMMAD TAUSEEF JAVED MBBS.DPH.FCPS(CM)M.Phil(CM).Dip-Card DR TAUSEEF JAVED
Statistics Measures of Central Tendency DR TAUSEEF JAVED
• 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
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
Mean (con’t) • Mean of a sample • Mean of a population DR TAUSEEF JAVED
Mean (con’t) Exam Scores 75 82 72 68 89 X sum all scores 91 78 94 88 75 n = total number of scores for the sample DR TAUSEEF JAVED
Mean (con’t) Performance and Memory S tudy 2 2 3 3 4 6 5 Frequency 4 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
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
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
Median (con’t) 2 Number of Words Recalled in Performance Study 2 3 3 4 4 4 4 4 10 DR TAUSEEF JAVED
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
DR TAUSEEF JAVED
Mode (con’t) 2 2 Number of Words Recalled in Performance Study 3 3 The mode is 4. 4 4 4 4 4 10 DR TAUSEEF JAVED
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
Red Blue Green Yellow Yellow Green Blue Red Green Red Yellow Blue DR TAUSEEF JAVED
Red Blue Green Yellow Yellow Green Blue Red Green Red Yellow Blue DR TAUSEEF JAVED
• 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
The best measure of central tendency depends on…. • The scale of measurement. • The shape of the distribution. DR TAUSEEF JAVED
Scales of Measurement • Nominal scale = mode • Ordinal scale = median • Interval scale = mean, median, or mode • Ratio scale = mean, median, or mode DR TAUSEEF JAVED
Shape of the Distribution Skew refers to the general shape of a distribution when it is graphed. Symmetrical = zero skew Scores clustered on the high or low end of a distribution = skewed distribution DR TAUSEEF JAVED
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 Scores The mean, median, and mode are the same. DR TAUSEEF JAVED
• The normal distribution is the “ideal” symmetrical distribution DR TAUSEEF JAVED
Distributions that are skewed have one side of the distribution where the data frequency tapers off DR TAUSEEF JAVED
Skewed Distribution Positive Skew 12 10 Frequency 8 6 4 2 0 27 32 37 42 47 52 57 62 67 72 77 Scores Tail points in the positive direction. DR TAUSEEF JAVED
Skewed Distribution Negative Skew 12 10 Frequency 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
The mean will either underestimate or overestimate the center of skewed distributions. Positive Skew Negative Skew 12 12 10 10 Frequency 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
Kurtosis • Measure of the flatness or peakedness of the distribution DR TAUSEEF JAVED

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Central tendency

  • 1. Measures of Central Tendency DR MUHAMMAD TAUSEEF JAVED MBBS.DPH.FCPS(CM)M.Phil(CM).Dip-Card DR TAUSEEF JAVED
  • 2. Statistics Measures of Central Tendency DR TAUSEEF JAVED
  • 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. 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. Mean (con’t) • Mean of a sample • Mean of a population DR TAUSEEF JAVED
  • 6. Mean (con’t) Exam Scores 75 82 72 68 89 X sum all scores 91 78 94 88 75 n = total number of scores for the sample DR TAUSEEF JAVED
  • 7. Mean (con’t) Performance and Memory S tudy 2 2 3 3 4 6 5 Frequency 4 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. 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. 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. Median (con’t) 2 Number of Words Recalled in Performance Study 2 3 3 4 4 4 4 4 10 DR TAUSEEF JAVED
  • 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
  • 13. Mode (con’t) 2 2 Number of Words Recalled in Performance Study 3 3 The mode is 4. 4 4 4 4 4 10 DR TAUSEEF JAVED
  • 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. Red Blue Green Yellow Yellow Green Blue Red Green Red Yellow Blue DR TAUSEEF JAVED
  • 16. Red Blue Green Yellow Yellow Green Blue Red Green Red Yellow Blue DR TAUSEEF JAVED
  • 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. The best measure of central tendency depends on…. • The scale of measurement. • The shape of the distribution. DR TAUSEEF JAVED
  • 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. Shape of the Distribution Skew refers to the general shape of a distribution when it is graphed. Symmetrical = zero skew Scores clustered on the high or low end of a distribution = skewed distribution DR TAUSEEF JAVED
  • 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 Scores The mean, median, and mode are the same. DR TAUSEEF JAVED
  • 22. • The normal distribution is the “ideal” symmetrical distribution DR TAUSEEF JAVED
  • 23. Distributions that are skewed have one side of the distribution where the data frequency tapers off DR TAUSEEF JAVED
  • 24. Skewed Distribution Positive Skew 12 10 Frequency 8 6 4 2 0 27 32 37 42 47 52 57 62 67 72 77 Scores Tail points in the positive direction. DR TAUSEEF JAVED
  • 25. Skewed Distribution Negative Skew 12 10 Frequency 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. The mean will either underestimate or overestimate the center of skewed distributions. Positive Skew Negative Skew 12 12 10 10 Frequency 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. Kurtosis • Measure of the flatness or peakedness of the distribution DR TAUSEEF JAVED