Biostatistics chapter two measure of the central
- 1. Measures of Central Tendency give us information about the location of the center of the distribution of data values. oA single value that describes the characteristics of the entire mass of data Objectives of measures of central tendency To get a single value that describe characteristics of the entire data To summarizing/reducing the volume of the data To facilitating comparison within one group or between groups of data Measures of Central Tendency 1
- 2. 2 Types of Measures of Central Tendency
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- 5. Example: calculate the mean for the following age distribution. Class frequency 6 - 10 35 11- 15 23 16- 20 15 21- 25 12 26- 30 9 31- 35 6 Solutions: • First find the class marks • Find the product of frequency and class marks • Find mean using the formula. 5
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- 7. Advantages of Arithmetic Mean: • is rigidly defined. • is based on all observation. • is suitable for further mathematical treatment. • is stable average, i.e. not affected by fluctuations of sampling to some extent. • It is easy to calculate and simple to understand. Disadvantages of Arithmetic Mean: • It is affected by extreme observations. • It can not be used in the case of open end classes. •It can not be determined by the method of inspection. • It can not be used when dealing with qualitative characteristics, such as intelligence, honesty, beauty. • It can be a number which does not exist in a serious. • Some times it leads to wrong conclusion if the details of the data from which it is obtained are not available. • It gives high weight to high extreme values and less weight to low extreme values. 7
- 10. 10 Values 2 4 6 8 10 frequency 1 2 2 2 1
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- 12. Properties of geometric mean a. Its calculations are not as such easy. b. It involves all observations during computation c. It may not be defined even it a single observation is negative d. If the value of one observation is zero its values becomes zero 12
- 13. 13 Harmonic Mean
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- 15. 15 The Median
- 16. Exercise: Find the median for the following data 16 The Median… x 6.7 6.2 8.4 6.9 7.1 f 5 2 8 7 7
- 17. 17 Median for Grouped Data.
- 18. 18 Example: Calculate the median for the following frequency distribution. Class 0-5 5-10 10-15 15-20 20-25 Frequency 5 8 10 8 5 less than cf. 5 13 23 31 36
- 19. The median is used to find the center or middle value of a data set. The median is used when it is necessary to find out whether the data values fall into the upper half or lower half of the distribution. The median is used for an open-ended distribution. The median is affected less than the mean by extremely high or extremely low values. 19 Properties of Median
- 20. 20 The Mode
- 21. 21 Mode for Grouped data
- 22. 22 Class 1.9-2.3 2.3-2.7 2.7-3.1 3.1-3.5 3.5-3.9 3.9-4.3 f 5 5 9 4 4 3
- 23. Advantages of Mode • It is not affected by extreme observations. • Easy to calculate and simple to understand. • It can be calculated for distribution with open end class Disadvantages of Mode • It is not rigidly defined. • It is not based on all observations • It is not suitable for further mathematical treatment. • It is not stable average, i.e. it is affected by fluctuations of sampling to some extent. • Often its value is not unique. 23
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- 25. 25 Example: find the quartiles for the following data. 1 2 4 6 2 2 4 5 6 3 4 2 6 7 8 9 Solution: First arrange the data according to increasing order 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6, 6, 7, 8, 9
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- 28. 28 Deciles
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- 30. 30 Percentiles
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- 33. Solutions: First find the less than cumulative frequency. Use the formula to calculate the required Quartiles Classes Frequency < Cf 140- 150 17 17 150- 160 29 46 160- 170 42 88 170- 180 72 160 180- 190 84 244 190- 200 107 351 200- 210 49 400 210- 220 34 434 220- 230 31 465 230- 240 16 481 240- 250 12 493 33
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- 37. It indicates that the degree to which numerical data tend to spread about an average value. The objective of measure of variation are: To control variability To compare two or more groups in terms of variability To make further statistical analysis The most commonly used measures of dispersions are: 1) Range 2) Interquartile range (IQR) 2) Mean deviation 3) Variance and Standard deviation 4) Coefficient of variation 37 MEASURE OF VARIATION (DISPERSION)
- 38. The Range The difference between the largest and smallest observations in a sample. Range = Maximum value – Minimum value Properties of range It is the simplest crude measure and can be easily understood It takes into account only two values which causes it to be a poor measure of dispersion Very sensitive to extreme observations The larger the sample size, the larger the range 38
- 39. 39 Interquartile range (IQR): Indicates the spread of the middle 50% of the observations, and used with median IQR = Q3 - Q1 Properties of IQR: It is a simple and versatile measure It encloses the central 50% of the observations It is not based on all observations but only on two specific values It is important in selecting cut-off points in the formulation of clinical standards Since it excludes the lowest and highest 25% values, it is not affected by extreme values Less sensitive to the size of the sample
- 40. 40 The Mean Deviation (M.D)
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- 42. 42 Steps to calculate M.D
- 43. 43 The deviations of each observation from mean, median and mode
- 44. 44 Population Variance and standard deviation
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- 47. Special properties of Standard deviations For normal (symmetric) distribution the following holds. • Approximately 68.27%, 95.45 % and 99.73% of the data values fall within one two and three standard deviation of the mean respectively . i.e. with in Chebyshev's Theorem • For any data set, the proportion of the values that fall with in k standard deviations of the mean or will be at least • standard deviations of the mean is at most
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- 50. 50 Exercise 1: The following data are the ages in years of 15 women who attend health education last year 30, 41, 39, 41, 32, 29, 35, 31, 30, 36, 33, 37, 32, 30, and 41. Calculate: A. Mean, Median and Mode B. IQR, MD, SD and CV C. All Quartile, 5th decile and the 20th percentile
- 51. 51 Exercise 2: Marks of 20 students are summarized in the following frequency distribution A. Mean, Median and Mode B. IQR, MD, SD and CV C. All Quartile, 5th decile and the 20th percentile
- 53. Skewness Skewness is the degree of asymmetry or departure from symmetry of a distribution. Skewness is concerned with the shape of the curve not size. Symmetrical skewed distribution: when the value is uniformly distributed around the mean (distribution of the data below the mean and above the mean are equal) i.e. Mean=Median = Mode Positively skewed distribution: if one or more observations are extremely large i.e. mean is greater than median and mode Negatively Skewed distribution: if one or more extremely small observations are present i.e. mean is smaller than median and mode. 53
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- 56. Kurtosis Kurtosis is a measure of the shape of a distribution relates to its flatness or peakednesss. Usually taken relative to a normal distribution. A distribution having relatively high peak is called leptokurtic. If a curve representing a distribution is flat topped, it is called Platykurtic. The normal distribution which is not very high peaked or flat topped is called mesokurtic. 56
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- 59. Thank you!