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# Mean, median, and mode ug

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• page 58 of text Name the two values if the set is bimodal
• Data skewed to the left is said to be ‘negatively skewed’ with the mean and median to the left of the mode. Data skewed to the right is said to be ‘positively skewed’ with the mean and media to the right of the mode.
• Data not ‘lopsided’.
• Data lopsided to left (or slants down to the left - definition of skew is ‘slanting’)
• Data lopsided to the right (or slants down to the right)
• Reminder: range is the highest score minus the lowest score
• Reminder: range is the highest score minus the lowest score
• Reminder: range is the highest score minus the lowest score
• These ideas will be used repeatedly throughout the course.
• page 79 of text
• Some student have difficulty understand the idea of ‘within one standard deviation of the mean’. Emphasize that this means the interval from one standard deviation below the mean to one standard deviation above the mean.
• These percentages will be verified by the concepts learned in Chapter 5. Emphasize the Empirical Rule is appropriate for data that is in a BELL-SHAPED distribution.
• ### Transcript

• 1. <ul><li>Single value in series of observations which indicate the characteristics of observations </li></ul><ul><li>All data / values clustered around it &amp; used to compare between one series to another </li></ul><ul><li>Measures: a) Mean ( Arithmetic / Geometric / Harmonic) </li></ul><ul><li>b) Median </li></ul><ul><li>c) Mode </li></ul>
• 2. <ul><li>  </li></ul><ul><li>It is sum of all observations divided by number of observations </li></ul><ul><li>  </li></ul><ul><li>__ Σx </li></ul><ul><li>Mean ( X ) = ------ ( x= observation &amp; n= no of observations) </li></ul><ul><li>n </li></ul><ul><li>  </li></ul><ul><li>  Problem : </li></ul><ul><li>  </li></ul><ul><li>ESR of seven subjects is 8,7, 9, 10, 7, 7 and 6. Calculate the mean. </li></ul><ul><li>  </li></ul><ul><li>8+7+9+10+7+7+6 54 </li></ul><ul><li>Mean= -------------------------- = ------- = 7.7 </li></ul><ul><li>7 7 </li></ul><ul><li>  </li></ul><ul><li>  </li></ul>
• 3. <ul><li>For discrete observation: </li></ul><ul><li>  If we have x 1 , x 2 , …… x n observations with corresponding frequencies f 1 , f 2 , ……f n, then </li></ul><ul><li>  x 1 f 1 + x 2 f 2 + ……. x n f n Σfx </li></ul><ul><li>Mean = --------------------------------- = ---------- </li></ul><ul><li>f 1 + f 2 + ……f n Σf </li></ul><ul><li>Problem : Calculate the avg. no. of children / family from the following data: </li></ul>Mean = 519/ 240 = 2.163 No. of Children (X) No. of families ( f ) Total no of children (fx) 0 30 0 x 30 = 0 1 52 1 x 52 = 52 2 60 2 x 60 = 120 3 65 3 x 65 = 195 4 18 4 x 18 = 72 5 10 5 x 10 = 50 6 5 6 x 5 = 30 Total = 240 = 519
• 4. <ul><li>When observations are arranged in ascending or descending order of magnitude, the </li></ul><ul><li>middle most value is known as Median </li></ul><ul><li>  </li></ul><ul><li>Problem : Same example of ESR as in mean </li></ul><ul><li>observations are arranged first in ascending order, i.e 6, 7, 7, 7, 8, 9, 10 </li></ul><ul><li>n+1 7+1 </li></ul><ul><li>When n is odd, Median = ------ th observation i.e, ------- = 4 th observation = 7 </li></ul><ul><li>2 2 </li></ul><ul><li>n/2 th + (n/2 +1) th observation </li></ul><ul><li>When n is even , Median = ---------------------------------------------- </li></ul><ul><li>2 </li></ul><ul><li>  So, if there are 8 observations of ESR like 5, 6, 7, 7,7, 8, 9, 10 </li></ul><ul><li>n/2 th + (n/2 +1) 4 th + 5 th 7+ 7 </li></ul><ul><li>Median = ------------------------th observation = ----------------th observation = --------- = 7 =   2 2 2 </li></ul>
• 5. <ul><li>The mode is the data item that appears the most. </li></ul><ul><li>If all data items appear the same number of times, then there is no mode. </li></ul>
• 6. 5, 4, 6, 11, 5, 7, 10, 5 The mode is 5.
• 7. <ul><li>Mode is 5 </li></ul><ul><li>Bimodal - 2 and 6 </li></ul><ul><li>No Mode </li></ul>a. 5 5 5 3 1 5 1 4 3 5 b. 1 2 2 2 3 4 5 6 6 6 7 9 c. 1 2 3 6 7 8 9 10 Examples
• 8. Merits Demerits <ul><li>Mean: </li></ul><ul><li>Rigidly defined </li></ul><ul><li>Based on all observations </li></ul><ul><li>Easy to calculate &amp; understand </li></ul><ul><li>Least affected by sampling fluctuation, </li></ul><ul><li>hence more stable </li></ul><ul><li>Mean: </li></ul><ul><li>Can be used only for quantitative data </li></ul><ul><li>Unduly affected by extreme observations </li></ul><ul><li>Median: </li></ul><ul><li>Not affected by extreme observations </li></ul><ul><li>Both for quantitative &amp; qualitative data </li></ul><ul><li>Median: </li></ul><ul><li>Affected more by sampling fluctuations </li></ul><ul><li>Not rigidly defined </li></ul><ul><li>Can be used for further mathematical calculation </li></ul><ul><li>Mode: </li></ul><ul><li>Not affected by extreme observations </li></ul><ul><li>Both for quantitative &amp; qualitative data </li></ul><ul><li>Mode: </li></ul><ul><li>Not rigidly defined </li></ul><ul><li>Can be used for further mathematical calculation </li></ul>
• 9. <ul><li>Symmetric </li></ul><ul><li>Data is symmetric if the left half of its histogram is roughly a mirror of its right half. </li></ul><ul><li>Skewed </li></ul><ul><li>Data is skewed if it is not symmetric and if it extends more to one side than the other. </li></ul>Definitions
• 10. Skewness Mode = Mean = Median SYMMETRIC Figure 2-13 (b)
• 11. Skewness Mode = Mean = Median SKEWED LEFT (negatively ) SYMMETRIC Mean Mode Median Figure 2-13 (b) Figure 2-13 (a)
• 12. Skewness Mode = Mean = Median SKEWED LEFT (negatively ) SYMMETRIC Mean Mode Median SKEWED RIGHT (positively) Mean Mode Median Figure 2-13 (b) Figure 2-13 (a) Figure 2-13 (c)
• 13. <ul><li>Biological variation in large groups is common. e.g : BP, wt </li></ul><ul><li>What is normal variation? and How to measure? </li></ul><ul><li>Measure of dispersion helps to find how individual observations are dispersed around the central tendency of a large series </li></ul><ul><li>Deviation = Observation - Mean </li></ul>
• 14. <ul><li>Range </li></ul><ul><li>Quartile deviation </li></ul><ul><li>Mean deviation </li></ul><ul><li>Standard deviation </li></ul><ul><li>Variance </li></ul><ul><li>Coefficient of variance : indicates relative variability ( SD/Mean) x100 </li></ul>
• 15. <ul><li>Range : difference between the highest and the lowest value </li></ul><ul><li>Problem: </li></ul><ul><li>Systolic and diastolic pressure of 10 medical students are as follows: 140/70, 120/88, 160/90, 140/80, 110/70, 90/60, 124/64, 100/62, 110/70 &amp; 154/90. Find out the range of systolic and diastolic blood pressure </li></ul><ul><li>Solution: </li></ul><ul><li>Range of systolic blood pressure of medical students: 90-160 or 70 </li></ul><ul><li>Range of diastolic blood pressure of medical students: 60-90 or 30 </li></ul><ul><li>Mean Deviation: average deviations of observations from mean value </li></ul><ul><li>_ </li></ul><ul><li>Σ (X – X ) __ </li></ul><ul><li>Mean deviation (M.D) = --------------- , ( where X = observation, X = Mean </li></ul><ul><li>n n= number of observation ) </li></ul>
• 16. <ul><li>  Problem : Find out the mean deviation of incubation period of measles of 7 children, which are as follows: 10, 9, 11, 7, 8, 9, 9. </li></ul><ul><li>Solution: </li></ul><ul><li>  </li></ul><ul><li>  </li></ul>Mean deviation (MD ) = _ Σ X - X = ------------ n = 6 / 7 = 0.85 Observation (X) __ Mean ( X ) __ Deviation (X - X) 10 __ X = Σ X / n = 63 / 7 = 9 1 9 0 11 2 7 -2 8 -1 9 0 9 0 ΣX=63 _ Σ (X-X) = 6, ignoring + or - signs
• 17. <ul><li>It is the most frequently used measure of dispersion </li></ul><ul><li>S.D is the Root-Means-Square-Deviation </li></ul><ul><li>S.D is denoted by σ or S.D </li></ul><ul><li>___________ </li></ul><ul><li>Σ ( X – X ) 2 </li></ul><ul><li>S.D (σ) = γ ---------------------- </li></ul><ul><li>n </li></ul>
• 18. <ul><li>Calculate the mean </li></ul><ul><li>↓ </li></ul><ul><li>Calculate difference between each observation and mean </li></ul><ul><li>↓ </li></ul><ul><li>Square the differences </li></ul><ul><li>↓ </li></ul><ul><li>Sum the squared values </li></ul><ul><li>↓ </li></ul><ul><li>Divide the sum of squares by the no. observations (n) to get ‘mean square deviation’ or variances (σ 2 ). [For sample size &lt; 30, it will be divided by (n-1)] </li></ul><ul><li>↓ </li></ul><ul><li>Find the square root of variance to get Root-Means-Square-Deviation or S.D ( σ) </li></ul>
• 19. S.D ( σ ) = = Σ(X –X) 2 / n-1 =(√1924/ (12-1) _____ = √174 = 13.2 Observation (X) __ Mean ( X ) _ Deviation (X- X) __ (X-X) 2 58 __ X = Σ X / n = 984/12 = 82 -12 576 66 -16 256 70 -12 144 74 -8 64 80 -2 4 86 -4 16 90 8 64 100 18 324 79 -3 9 96 14 196 88 6 36 97 15 225 Σ X = 984 _ Σ (X - X) 2 =1914
• 20. Estimation of Standard Deviation Range Rule of Thumb x - 2 s x x + 2 s Range  4 s or (minimum usual value) (maximum usual value)
• 21. Estimation of Standard Deviation Range Rule of Thumb x - 2 s x x + 2 s Range  4 s or (minimum usual value) (maximum usual value) Range 4 s 
• 22. Estimation of Standard Deviation Range Rule of Thumb x - 2 s x x + 2 s Range  4 s or (minimum usual value) (maximum usual value) Range 4 s  = highest value - lowest value 4
• 23. &nbsp;
• 24. <ul><li>minimum ‘usual’ value  (mean) - 2 (standard deviation) </li></ul><ul><li>minimum  x - 2(s) </li></ul>
• 25. <ul><li>minimum ‘usual’ value  (mean) - 2 (standard deviation) </li></ul><ul><li>minimum  x - 2(s) </li></ul><ul><li>maximum ‘usual’ value  (mean) + 2 (standard deviation) </li></ul><ul><li>maximum  x + 2(s) </li></ul>
• 26. x The Empirical Rule (applies to bell-shaped distributions ) FIGURE 2-15
• 27. x - s x x + s 68% within 1 standard deviation 34% 34% The Empirical Rule (applies to bell-shaped distributions ) FIGURE 2-15
• 28. x - 2s x - s x x + 2s x + s 68% within 1 standard deviation 34% 34% 95% within 2 standard deviations The Empirical Rule (applies to bell-shaped distributions ) 13.5% 13.5% FIGURE 2-15
• 29. x - 3s x - 2s x - s x x + 2s x + 3s x + s 68% within 1 standard deviation 34% 34% 95% within 2 standard deviations 99.7% of data are within 3 standard deviations of the mean The Empirical Rule (applies to bell-shaped distributions ) 0.1% 2.4% 2.4% 13.5% 13.5% FIGURE 2-15 0.1%