Descriptive Analysis in Statistics
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This document describes various methods of descriptive data analysis, including: 1) Descriptive analysis uses measures of central tendency like mean, median, and mode to summarize large datasets. It is used to describe data through tables and diagrams. 2) Variables can be categorized as dependent or independent. Descriptive analysis examines relationships between these types of variables. 3) Common descriptive methods include frequency distributions, measures of central tendency and variability, as well as graphical presentations like histograms, bar charts, and scatter plots. These methods organize and illustrate key characteristics of quantitative data.
Descriptive Analysis in Statistics
- 1. Medicine & Society II Descriptive Analysis Dr Azmi Mohd Tamil Dept of Community Health Universiti Kebangsaan Malaysia
- 3. Dependent/Independent Independent Variables Food Intake Frequency of Exercise Obesity Dependent Variable
- 5. Data Analysis Descriptive Bivariate Multivariate
- 6. Descriptive Summarise a large set of data by a few meaningful numbers. For the purpose of describing the data Example; in one year, what kind of cases are treated by the Psychiatric Dept? Tables & diagrams are usually used to describe the data For numerical data, measures of central tendency & spread is usually used
- 7. Frequency Table Race F % Malay 760 95.84% Chinese 5 0.63% Indian 0 0.00% Others 28 3.53% TOTAL 793 100.00% •Illustrates the frequency observed for each category
- 8. Disease Prevalence: Hypertension Of those previously 140 diagnosed as 120 hypertensive; 100 Only 26% have normal 80 Normal BP 60 Brdrline 27.1% borderline Hiprtnsi 40 46.9% hypertensive 20 0 BP
- 9. Frequency Distribution Table • > 20 observations, best Umur Bil % presented as a frequency 0-0.99 25 3.26% 1-4.99 78 10.18% distribution table. 5-14.99 140 18.28% •Columns divided into class & 15-24.99 126 16.45% 25-34.99 112 14.62% frequency. 35-44.99 90 11.75% 45-54.99 66 8.62% •Mod class can be determined 55-64.99 60 7.83% using such tables. 65-74.99 50 6.53% 75-84.99 16 2.09% 85+ 3 0.39% JUMLAH 766
- 10. Measurement of Central Tendency & Spread
- 11. Measures of Central Tendency Mean Mode Median
- 12. Variability Standard deviation Inter-quartiles Skewness & kurtosis
- 13. Mean the average of the data collected To calculate the mean, add up the observed values and divide by the number of them. A major disadvantage of the mean is that it is sensitive to outlying points
- 14. Mean: Example 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Total of x = 648 n= 20 Mean = 648/20 = 32.4
- 15. Measures of variation - standard deviation tells us how much all the scores in a dataset cluster around the mean. A large sd is indicative of a more varied data scores. a summary measure of the differences of each observation from the mean. If the differences themselves were added up, the positive would exactly balance the negative and so their sum would be zero. Consequently the squares of the differences are added.
- 16. sd: Example x (x-mean)^2 x (x-mean)^2 12, 13, 17, 21, 24, 24, 12 416.16 32 0.16 26, 27, 27, 30, 32, 35, 13 376.36 35 6.76 37, 38, 41, 43, 44, 46, 17 237.16 37 21.16 53, 58 21 129.96 38 31.36 24 70.56 41 73.96 Mean = 32.4; n = 20 24 70.56 43 112.36 Total of (x-mean)2 26 40.96 44 134.56 = 3050.8 27 29.16 46 184.96 27 29.16 53 424.36 Variance = 3050.8/19 30 5.76 58 655.36 = 160.5684 TOTAL 1405.8 TOTAL 1645 sd = 160.56840.5=12.67
- 18. Median the ranked value that lies in the middle of the data the point which has the property that half the data are greater than it, and half the data are less than it. if n is even, average the n/2th largest and the n/2 + 1th largest observations "robust" to outliers
- 19. Median: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 (20+1)/2 = 10th which is 30, 11th is 32 Therefore median is (30 + 32)/2 = 31
- 20. Measures of variation - quartiles The range is very susceptible to what are known as outliers A more robust approach is to divide the distribution of the data into four, and find the points below which are 25%, 50% and 75% of the distribution. These are known as quartiles, and the median is the second quartile.
- 21. Quartiles 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 25th percentile 24; (24+24)/2 50th percentile 31; (30+32)/2 75th percentile 42.5; (41+43)/2
- 22. Mode The most frequent occurring number. E.g. 3, 13, 13, 20, 22, 25: mode = 13. It is usually more informative to quote the mode accompanied by the percentage of times it happened; e.g, the mode is 13 with 33% of the occurrences.
- 23. Mode: Example 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Modes are 24 (10%) & 27 (10%)
- 24. Mean or Median? Which measure of central tendency should we use? if the distribution is normal, the mean will be the measure to be presented, otherwise the median should be more appropriate.
- 26. Presentation Qualitative & Quantitative Data Charts & Tables
- 28. Graphing Categorical Data: Univariate Data Categorical Data Graphing Data Tabulating Data The Summary Table Pie Charts CD S avings B onds Bar Charts Pareto Diagram S toc k s 45 120 40 100 0 10 20 30 40 50 35 30 80 25 60 20 15 40 10 20 5 0 0 S toc k s B onds S avings CD
- 29. Bar Chart 80 69 60 40 20 20 Percent 11 0 Housew ife Office w ork Field w ork Type of work
- 30. Pie Chart Others Chinese Malay
- 31. Tabulating and Graphing Bivariate Categorical Data Contingency tables: Table 1: Contigency table of pregnancy induced hypertension and SGA Count SGA Normal SGA Total Pregnancy induced No 103 94 197 hypertension Yes 5 16 21 Total 108 110 218
- 32. Tabulating and Graphing Bivariate Categorical Data 120 Side by 100 103 94 side 80 charts 60 40 SGA 20 Normal Count 16 0 SGA No Yes Pregnancy induced hypertension
- 34. Tabulating and Graphing Numerical Data Numerical Data 41, 24, 32, 26, 27, 27, 30, 24, 38, 21 Frequency Distributions Ordered Array Ogive Cumulative Distributions 120 21, 24, 24, 26, 27, 27, 30, 32, 38, 41 100 80 60 40 20 0 2 144677 Area Stem and Leaf Histograms 10 20 30 40 50 6 Display 3 028 7 6 4 1 5 Tables Polygons 4 3 2 1 0 10 20 30 40 50 60
- 35. Tabulating Numerical Data: Frequency Distributions Sort raw data in ascending order: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Find range: 58 - 12 = 46 Select number of classes: 5 (usually between 5 and 15) Compute class interval (width): 10 (46/5 then round up) Determine class limits: 10.0-19.9, 20.0-29.9, 30.0-39.9 etc Determine class boundaries: e.g. (19.9+20.0)/2=19.95 Compute class midpoints: e.g. (10+19.9)/2 = 14.95 Count observations & assign to classes (i.e. use tally method)
- 36. Frequency Distributions and Percentage Distributions Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Class Midpoint Freq % 10.0 - 19.9 14.95 3 15% 20.0 - 29.9 24.95 6 30% 30.0 - 39.9 34.95 5 25% 40.0 - 49.9 44.95 4 20% 50.0 - 59.9 54.95 2 10% TOTAL 20 100%
- 37. Graphing Numerical Data: The Histogram Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 7 6 6 5 5 Frequency 4 4 3 No Gaps Between 3 2 2 Bars 1 0 14.95 24.95 34.95 44.95 54.95 Age Class Boundaries Class Midpoints
- 38. Graphing Numerical Data: The Frequency Polygon Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 7 6 5 4 3 2 1 0 14.95 24.95 34.95 44.95 54.95 Class Midpoints
- 39. Calculate Measures of Central Tendency & Spread We can use frequency distribution table to calculate; • Mean • Standard Deviation • Median • Mode
- 40. Mean Class Midpoint Freq freq x m.p. Mean = 659/20 10.0 - 19.9 14.95 3 44.85 = 32.95 20.0 - 29.9 24.95 6 149.70 Compare with 32.4 30.0 - 39.9 34.95 5 174.75 from direct 40.0 - 49.9 44.95 4 179.80 calculation. 50.0 - 59.9 54.95 2 109.90 TOTAL 20 659.00
- 41. Standard deviation Mid Class Point Freq f.m.p. f.mp^2 14.95 3 44.85 s2=((24634.05-(6592/20))/19) 10.0 - 19.9 670.51 s2=2920.05/19 20.0 - 29.9 24.95 6 149.70 3735.02 s2=153.69 30.0 - 39.9 34.95 5 174.75 6107.51 s = 12.4 40.0 - 49.9 44.95 4 179.80 8082.01 Compare with 12.67 from direct measurement. 50.0 - 59.9 54.95 2 109.90 6039.01 TOTAL 20 659.00 24634.05
- 42. Median Class Freq L1 +i *((n+1)/2) – f1 fmed 10.0 - 19.9 3 f1 = cumulative freq above median class 20.0 - 29.9 6 29.95 + 10((21/2)-9) 30.0 - 39.9 5 median class 5 29.95 + 15/5 = 32.95 40.0 - 49.9 4 From direct calculation, median = 31 50.0 - 59.9 2 TOTAL 20
- 43. Mode =L1 +i *(Beza1/(Beza1+Beza2)) Class Freq =19.95 + 10(3/(3+1)) =27.45 10.0 - 19.9 3 20.0 - 29.9 6 mode class Compare with modes of 24 & 27 30.0 - 39.9 5 from direct 40.0 - 49.9 4 calculation. 50.0 - 59.9 2 TOTAL 20
- 44. Graphing Bivariate Numerical Data (Scatter Plot ) 5.0 4.5 4.0 3.5 3.0 2.5 Birth weight 2.0 1.5 Rsq = 0.2028 30 40 50 60 70 80 90 100 Weight at first ANC
- 45. Principles of Graphical Excellence Presents data in a way that provides substance, statistics and design Communicates complex ideas with clarity, precision and efficiency Gives the largest number of ideas in the most efficient manner Almost always involves several dimensions Tells the truth about the data
- 46. Errors in Presenting Data Using “chart junk” Failing to provide a relative basis in comparing data between groups Compressing the vertical axis Providing no zero point on the vertical axis
- 47. “Chart Junk” Bad Presentation Minimum charge Good Presentation per visit Minimum charge 1960: $1.00 $ per visit 4 1970: $1.60 2 1980: $3.10 0 1990: $3.80 1960 1970 1980 1990
- 48. No Relative Basis Bad Presentation Good Presentation A’s received by A’s received by Freq. students. students. 300 30 % 200 20 100 10 0 0 Yr1 Yr2 Yr3 Yr4 Yr1 Yr2 Yr3 Yr4
- 49. Compressing Vertical Axis Bad Presentation Good Presentation HUKM Quarterly HUKM Quarterly $ Profits $ Profits 200 50 100 25 0 0 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4
- 50. No Zero Point on Vertical Axis Bad Presentation Good Presentation HUKM Monthly HUKM Monthly $ Collection $ Collection 45 45 42 42 39 39 36 36 J F M A M J 0 J F M A M J Graphing the first six months of collection.