Statistical methods

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central of tendency and measure of dispersion.

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Statistical methods

  1. 1. March 2012 Statistcal methods
  2. 2. Budget Procedure The Budget Will Be Shown As A Quality Process As The Slides Will Be Divided According To The 5 Steps Of  Say What You Do Quality.  Do What You Say  Record What You Do  Review What You Do  Restart The Process
  3. 3. Say What You Do (Contents)The Budget Shall Consist The Following Parameters Need To Describe Central Tendency Types Of Central Tendencies Comparing The 3 tendencies Skewness Of Distribution Need To Measure Dispersion
  4. 4. Do What You Say &Record What You Do Both Steps Are CollaboratedBecause recording of the Processes shall be done side by side so as to find the mistakes ASAP……… And Here We Present The Budget
  5. 5. Why Describe Central Tendency? Data often cluster around a central value that lies between the two extremes. This single number can describe the value of scores in the entire data set. There are three measures of central tendency. 1) Mean 2) Median 3) Mode
  6. 6. The Mode The mode is the most frequently occurring number in a set of data. • E.g., Find the mode of the following numbers… • 15, 20, 21, 23, 23, 23, 25, 27, 30 Also, if there are two modes, the data set is bimodal. If there are more than two modes, the data set is said to be multimodal.
  7. 7. The Median The middle score when all scores in the data set are arranged in order. Half the scores lie above and half lie below the median. E.g., Find the median of the following numbers… 10, 12, 14, 15, 17, 18, 20.
  8. 8.  When there are an even number of scores, you must take the average of the middle two scores. Eg., 10, 12, 14, 15, 17, 18 (14 + 15)/2 = 14.5.
  9. 9.  The median can also be calculated from a frequency distribution.  E.g., A stats class received the following marks out of 20 on their first exam.X freq Cumulativefreq20 1 1519 2 1416 2 1214 1 10 What is the median grade?12 4 911 2 510 3 3
  10. 10.  Step 1 - Multiply 0.5 times N + 1 to obtain the location of the middle frequency. 0.5(15 + 1) = 8 Step 2 - Locate this score on your frequency distribution. 12
  11. 11. The Mean This is the sum of all the scores data set divided by the number of scores in the set. E.g., What’s the mean of the ∑x following test scores?x = n 56, 65, 75, 83, 92 x = 371/5 = 74.2
  12. 12.  The mean can also be calculated using a frequency distribution. The following scores were obtained on a stats exam marked out of 20. X freq 20 1 19 2 16 2 Find the mean of the exam 14 1 12 4 scores. 11 2 10 3
  13. 13.  Multiply each score by the frequency. Add them together and divide by NX freq fX20 1 20 X = X fX/N19 2 3816 2 3214 1 14 = 204/1512 4 4811 2 22 = 13.610 3 30 N = 15 NfX = 204
  14. 14. Characteristics of the Mean Summed deviations about the mean equal 0.Score X-X 2 2 - 5 = -3 3 3 - 5 = -2 5 5-5=0 7 7-5=2__8__ 8-5=3_ X = 25 8 (x - x) = 0X=5
  15. 15.  The mean is sensitive to extreme scores. Score Score Note, the median 2 2 remains the same in 3 3 both cases. 5 5 7 7 __8__ __33__ _ X = 25 _ X = 50 X=5 X = 10
  16. 16.  The sum of squared deviations is least about the mean Score (X - X)2 2 (2 - 5)2 = 9 3 (3 - 5)2 = 4 5 (5 - 5)2 = 0 7 (7 - 5)2 = 4 __8__ (8 - 5)2 = 9 _ X = 25 (x - x)2 = 26 X=5
  17. 17. Comparison of the Mean,Median, and Mode The mode is the roughest measure of central tendency and is rarely used in behavioral statistics. Mean and median are generally more appropriate. If a distribution is skewed, the mean is pulled in the direction of the skew. In such cases, the median is a better measure of central tendency.
  18. 18. Skewness of Distribution  Comparing the mean and the median Normal Negative Positive Skew SkewDistribution Mean & Median Mean Mean MedianMedian the same
  19. 19. Why Measure Dispersion? Measures of dispersion tell us how spread out the scores in a data set are. Surely all scores will not be equal to the mean. There are four measures of dispersion we will look at: • Range (crude range) • Standard Deviation
  20. 20. The Range  The simplest measure of variability. Simply the highest score minus the lowest score.  Limited by extreme scores or outliers.E.g., Find the range in the following test scores. 100, 74, 68, 68, 57, 56 Range = H - L = 100 - 56 = 44
  21. 21. The Variance The sum of the squared deviations from the mean divided by N. ∑ (x - x) 2 s 2 = N
  22. 22. Calculating Variance (Deviation Formula) X X-X (X -X)2 12 3 9 11 2 4 10 1 1 9 0 0 9 0 0 9 0 0 8 -1 1 7 -2 4 6 -3 9 ∑ x = 81 ∑ (x - x) = 0 ∑ (x - x)2 = 28 x=9 S2 = ∑ (x - x)2 = 28 = 3.11 n 9
  23. 23. Calculating StandardDeviation Simply calculate the square root of the variance. So if s2 from the previous example was 3.11, the standard deviation (denoted by s) is 1.76.
  24. 24. Calculating the Variance and/orStandard Deviation Formulae: Variance: Standard Deviation:s 2 = ∑( X − X ) i 2 s= ∑( X − X ) i 2 N N Examples Are As Follows
  25. 25. Example: Data: X = {6, 10, 5, 4, 9, 8}; N=6 Mean: X X−X (X − X ) 2 X= ∑X = 42 =7 6 -1 1 N 6 10 3 3 9 Variance: 5 -2 4 S2 = ∑ (x - x)2 = 28 = 4.67 n 6 4 -3 9 9 2 2 4 Standard Deviation: 8 1 1 1 s = s 2 = 4.67 = 2.16Total: 42 Total: 28
  26. 26. Review What You Do Need To Describe Central Tendency Types Of Central Tendencies Comparing The 3 tendencies Skewness Of Distribution Need To Measure Dispersion
  27. 27. Do We Pass The Quality Test? No Or Yes
  28. 28. Quality Not AchievedPlease tell where we lacked and were wrong.
  29. 29. The Process Shall StartAgain
  30. 30. Budget Ends
  31. 31. Quality Achieved
  32. 32. Budget Ends

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