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2 descriptive statistics
- 1. Descriptive Statistics Maria Sharif Biostatistician Departmentof public health
- 2. Outline • Definition ofDescriptive Statistics • Measures of Central Tendency • Mean • Median • Mode • Measures of Dispersion • The Range • IQR (Inter-Quartile Range) • Variance • Standard Deviation • Introduction to Normal Distribution 4/22/2021 Descriptive Statistics 2
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- 4. What is ameasure of Central Tendency? • Numbers that describe what is average or typical of the distribution • You can think of this value as where the middle of a distribution lies. 4/22/2021 Descriptive Statistics 4
- 5. The Mode • Thecategory or score with the largest frequency (or percentage) in the distribution. • The mode can be calculated for variables with levels of measurement that are: nominal, ordinal, or discrete quantitative. 4/22/2021 Descriptive Statistics 5
- 6. The Mode: AnExample • Example: Number of Votes for Candidates for Mayor. The mode, in this case, gives you the “central” response of the voters: the most popular candidate. Candidate A – 11,769 votes The Mode: Candidate B – 39,443 votes “Candidate C” Candidate C – 78,331 votes 4/22/2021 Descriptive Statistics 6
- 7. The Median • Thescore that divides the distribution into two equal parts, so that half the cases are above it and half below it. • The median is the middle score, or average of middle scores in a distribution. 4/22/2021 Descriptive Statistics 7
- 8. Median Exercise #1(N is odd) Calculate the median for this hypothetical distribution: Job Satisfaction Frequency Very High 2 High 3 Moderate 5 Low 7 Very Low 4 TOTAL 21 4/22/2021 Descriptive Statistics 8
- 9. Median Exercise #2(N is even) Calculate the median for this hypothetical distribution: Satisfaction with Health Frequency Very High 5 High 7 Moderate 6 Low 7 Very Low 3 TOTAL 28 4/22/2021 Descriptive Statistics 9
- 10. Finding the Medianin Grouped Data w f Cf N L Median ) 5 (. 4/22/2021 Descriptive Statistics 10
- 11. The Mean • Thearithmetic average obtained by adding up all the scores and dividing by the total number of scores. 4/22/2021 Descriptive Statistics 11
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- 13. Formula for theMean N Y Y “Y bar” equals the sum of all the scores, Y, divided by the number of scores, N. 4/22/2021 Descriptive Statistics 13
- 14. Calculating the meanwith grouped scores where: f Y = a score multiplied by its frequency N Y f Y 4/22/2021 Descriptive Statistics 14
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- 18. Grouped Data: theMean & Median Number of People Age 18 or older living in a U.S. Household in 1996 (GSS 1996) Number of People Frequency 1 190 2 316 3 54 4 17 5 2 6 2 TOTAL 581 Calculate the median and mean for the grouped frequency below. 4/22/2021 Descriptive Statistics 18
- 19. Shape of theDistribution • Symmetrical (mean is about equal to median) • Skewed • Negatively (example: years of education) mean < median • Positively (example: income) mean > median • Bimodal (two distinct modes) • Multi-modal (more than 2 distinct modes) 4/22/2021 Descriptive Statistics 19
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- 21. Considerations for Choosinga Measure of Central Tendency • For a nominal variable, the mode is the only measure that can be used. • For ordinal variables, the mode and the median may be used. The median provides more information (taking into account the ranking of categories.) • For interval-ratio variables, the mode, median, and mean may all be calculated. The mean provides the most information about the distribution, but the median is preferred if the distribution is skewed. 4/22/2021 Descriptive Statistics 21
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- 24. The Importance of MeasuringVariability • Central tendency - Numbers that describe what is typical or average (central) in a distribution • Measures of Variability - Numbers that describe diversity or variability in the distribution. These two types of measures together help us to sum up a distribution of scores without looking at each and every score. Measures of central tendency tell you about typical (or central) scores. Measures of variation reveal how far from the typical or central score that the distribution tends to vary. 4/22/2021 Descriptive Statistics 24
- 25. Notice that bothdistributions have the same mean, yet they are shaped differently 4/22/2021 Descriptive Statistics 25
- 26. The Range Range =highest score - lowest score • Range – A measure of variation in interval-ratio variables. It is the difference between the highest (maximum) and the lowest (minimum) scores in the distribution. 4/22/2021 Descriptive Statistics 26
- 27. Inter-Quartile Range • Inter-QuartileRange (IQR) – A measure of variation for interval-ratio data. It indicates the width of the middle 50 percent of the distribution and is defined as the difference between the lower and upper quartiles (Q1 and Q3.) • IQR = Q3 – Q1 4/22/2021 Descriptive Statistics 27
- 28. The difference betweenthe Range and IQR Shows greater variability These values fall together closely Yet the ranges are equal! Importance of the IQR 4/22/2021 Descriptive Statistics 28
- 29. Variance • Variance –A measure of variation for interval-ratio variables; it is the average of the squared deviations from the mean 1 ) ( 2 2 N s Y Y Y 4/22/2021 Descriptive Statistics 29
- 30. Standard Deviation • StandardDeviation – A measure of variation for interval-ratio variables; it is equal to the square root of the variance. 1 ) ( 2 2 N s s Y Y Y 4/22/2021 Descriptive Statistics 30
- 31. Find the Meanand the Standard Deviation 4/22/2021 Descriptive Statistics 31
- 32. Considerations for Choosinga Measure of Variability • For nominal variables, you can only use IQV (Index of Qualitative Variation.) • For ordinal variables, you can calculate the IQV or the IQR (Inter-Quartile Range.) Though, the IQR provides more information about the variable. • For interval-ratio variables, you can use IQV, IQR, or variance/standard deviation. The standard deviation (also variance) provides the most information, since it uses all of the values in the distribution in its calculation. 4/22/2021 Descriptive Statistics 32
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- 34. Normal Distribution Why arenormal distributions so important? • Many dependent variables are commonly assumed to be normally distributed in the population • If a variable is approximately normally distributed we can make inferences about values of that variable • Example: Sampling distribution of the mean 4/22/2021 Descriptive Statistics 34
- 35. Normal Distribution • Symmetrical,bell-shaped curve • Also known as Gaussian distribution • Point of inflection = 1 standard deviation from mean • Mathematical formula f(X) 1 2 (e) (X )2 2 2 4/22/2021 Descriptive Statistics 35
- 36. • Since weknow the shape of the curve, we can calculate the area under the curve • The percentage of that area can be used to determine the probability that a given value could be pulled from a given distribution • The area under the curve tells us about the probability- in other words we can obtain a p-value for our result (data) by treating it as a normally distributed data set. 4/22/2021 Descriptive Statistics 36
- 37. Key Areas underthe Curve • For normal distributions + 1 SD ~ 68% + 2 SD ~ 95% + 3 SD ~ 99.9% 4/22/2021 Descriptive Statistics 37
- 38. Example IQ mean= 100 s = 15 4/22/2021 Descriptive Statistics 38
- 39. Normal Probability Distributions StandardNormalDistribution–N(0,1) • Weagree to use the standard normal distribution • Bell shaped • =0 • =1 • Note: not all bell shaped distributions are normal distributions 4/22/2021 Descriptive Statistics 39
- 40. Normal Probability Distribution • Cantake on an infinite number of possible values. • The probability of any one of those values occurring is essentially zero. • Curve has area or probability = 1 4/22/2021 Descriptive Statistics 40
- 41. What you havelearned today 1. Calculate the mean, median, mode 2. Calculate standard deviation, variance, IQR 3. Know when to use each 4. Understand what normal distribution is 4/22/2021 Descriptive Statistics 41
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