Introduction biostatistics
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This document introduces common measures of central tendency (mean, median, mode) and variation (range, variance, standard deviation, coefficient of variation) in biostatistics. It defines each measure and provides examples of calculating and interpreting them. The mean is the most common measure of central tendency but the median is more robust to outliers. The choice of central tendency measure depends on whether the data is skewed. The mode is used for measuring popularity. Measures of variation quantify how spread out the data values are.
Introduction biostatistics
- 1. Introduction to Biostatistics DR. SYED SANOWAR ALI
- 2. CENTRAL TENDENCY The centre of the distribution Or The most typical case
- 3. Measures of CENTRAL TENDENCY Given a data set, a measure of the CENTRAL TENDENCY is a value about which the observations tend to cluster In other wordsIn other words a measure of the CENTRAL TENDENCY is a value around whichCENTRAL TENDENCY is a value around which a data set is centereda data set is centered
- 4. Measures of CENTRAL TENDENCY The three most common measures are • Mean • Median • Mode
- 5. Mean: It is the value that is closest to all the other values in a distribution.
- 6. Mean = X1 + X2 + -------- Xn or n µ = X1 + X2 + -------- XN or N ∑ = summation = X bar µ = mu N = total number of values in population n = total number of values in sample × × n x x ∑ = N x∑ =µ
- 7. Find the mean of the following five salaries 6000, 10000, 14000, 50000, 10000 • Step 1. Arrange the values in ascending order. 6000, 10000, 10000, 14000, 50000 • Step 2. Add all of the observed values in the distribution. 6000+10000+10000+14000+50000= 90000 • Step 3. Divide the sum by the number of observations. 90000 / 5 = 18000 • Therefore, the mean salary is 18000 n x x ∑ =
- 8. Properties of Mean 1. One computes the mean by using all the values of the data. 2. The mean is used in computing other statistics, such as variance 3. The mean for the data set is unique and not necessarily one of the data value 4. The mean is affected by extremely high or low values, called outliers, and may not be the appropriate to use in these situation
- 9. Median is the middle value of a set of data that has been put into rank order. The median is also the 50th percentile of the distribution. Median
- 10. Example A: Odd Number of Observations Find the median of the following 6000, 10000, 14000, 50000, 10000 • Step 1. Arrange the values in ascending order. 6000, 10000, 10000, 14000, 50000 • Step 2. Find the middle position of the distribution by using (n + 1) / 2. Middle position = (5 + 1) / 2 = 6 / 2 = 3 • Therefore, the median will be the value at the third observation. • Step 3. Identify the value at the middle position. Third observation = 10000
- 11. Example A: Even Number of Observations Find the median of the following 6000, 10000, 14000, 50000, 10000, 12000 • Step 1. Arrange the values in ascending order. 6000, 10000, 10000, 12000, 14000, 50000 • Step 2. Find the middle position of the distribution by using (n + 1) / 2. Middle position = (6 + 1) / 2 = 7 / 2 = 3.5 • Step 3. Identify the value at the middle position. The median equals the average of the values of the third (value = 10000) and fourth (value = 12000 observations: Median = (10000 + 12000) / 2 = 11000
- 12. Properties of Median 1. The median is used when one must find the center or middle value 2. The median is used when one must determine whether the data values fall into the upper half or lower half of the distribution 3. The median is affected less than mean by extremely high or extremely low values
- 13. Mode is the value that occurs most often in a set of data. It can be determined simply by tallying the number of times each value occurs.
- 14. Mode In this case salary 10000 is the value that occurs most frequently. The mode is 10000 It should be noted that there can be more than one mode for a data set
- 15. Properties of Mode 1. The mode is used when the most typical case is desired 2. The mode is the easiest to compute 3. The mode can be used when the data are nominal such as religious preference, gender, or political affiliation 4. The mode is not always unique. A data set can have more than one mode, or the mode may not exist for a data set
- 16. Find the mean of the following incubation periods for hepatitis A: 27, 31, 15, 30, and 22 days. • Step 1. Arrange the values in ascending order distribution. 15, 22, 27, 30, 31 Step 2. Add all of the observed values in the distribution. 15 + 22 + 27 + 30 + 31 = 125 • Step 3. Divide the sum by the number of observations. 125 / 5 = 25.0 • Therefore, the mean incubation period is 25.0 days.
- 17. Example B: Even Number of Observations Suppose a sixth case of hepatitis was reported. hepatitis A: 27, 31, 15, 30, 22 and 29 days. • Step 1. Arrange the values in ascending order. 15, 22, 27, 29, 30, and 31 days • Step 2. Find the middle position of the distribution by using (n + 1) / 2. Middle location = 6 + 1 / 2 = 7 / 2 = 3½ • Step 3. Identify the value at the middle position. The median equals the average of the values of the third (value = 27) and fourth (value = 29) observations: Median = (27 + 29) / 2 = 28 days
- 18. Example B: Find the mode of the following incubation periods for hepatitis A: 27, 31, 15, 30, and 22 days. • Step 1. Arrange the values in ascending order. 15, 22, 27, 30, and 31 days • Step 2. Identify the value that occurs most often. None • Note: When no value occurs more than once, the distribution is said to have no mode.
- 19. the number of doses of diphtheria-pertussis- tetanus (DPT) vaccine each of seventeen 2-year-old children in a particular village received: 0, 0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4 Two children received no doses; two children received 1 dose; three received 2 doses; six received 3 doses; and four received all 4 doses. Therefore, the mode is 3 doses, because more children received 3 doses than any other number of doses.
- 20. Which measure of CT should you use ? The Mean is by far the most common measure of CT. It uses all of the information in the sample. This measure is very good when the distribution is symmetrical.
- 21. Mean , Median and Mode Data: 4000, 4500, 5000, 5500, 6000, 6000, 6500, 7000, 7500 and 8000 Mean = 6000 Median = 6000 Mode = 6000 = Same= Same
- 22. Salary Mean , Median and Mode= SameMean , Median and Mode= Same Normal Distribution Or Curve
- 23. Which measure of CT should you use ? If the distribution is skewed or there are extreme values the Mean is artificially pulled towards the extreme value. Age example: 19, 20, 21, 22, 49 Mean=26.2 yrs.Mean=26.2 yrs. Mean=49.2Mean=49.2 Marks example 05, 55, 57, 63, 66
- 24. Which measure of CT should you use ? Age : 19, 20, 21, 22, 49 Mean=26.2 yrs.Mean=26.2 yrs. Right skewed or Positively skewed
- 26. Which measure of CT should you use ? Marks 05, 55, 57, 63, 66 Mean=49.2Mean=49.2 Left skewed or Negatively skewed
- 28. Which measure of CT should you use ? • If the distribution is skewed or there are extreme values, in such a case Median proves to be better measure of the CT. • Median is resistant to extreme observations.
- 29. Which measure of CT should you use ? • Mode is commonly used as a measure of popularity that reflect CT of Opinion • Examples: 1. Most preferred pain killer 2. Most preferred model of washing machine 3. Most popular candidate
- 30. Most fighting cricket team • Pakistan=1 • Australia=2 • India=3 • England=4 1, 2, 4, 1, 2, 1, 3, 1, 4, 1,1, 2, 4, 1, 2, 1, 3, 1, 4, 1, 2, 1, 3, 2, 4, 4, 1, 1, 1, 4,2, 1, 3, 2, 4, 4, 1, 1, 1, 4, 3, 1, 1, 4, 2, 1, 1, 2, 1, 2,3, 1, 1, 4, 2, 1, 1, 2, 1, 2, 1, 4, 1, 1, 3, 2, 4, 1, 4, 11, 4, 1, 1, 3, 2, 4, 1, 4, 1 Which measure of CT should you use ?Mean(2.075) MODE 19 88 44 99 Median(2) Mode(1)
- 31. Measurement of Variation Measurement of Dispersion OR
- 32. Range The range is the simplest measure of variation to find. It is simply the highest value minus the lowest value. RANGE = MAXIMUM - MINIMUM Since the range only uses the largest and smallest values, it is greatly affected by extreme values, that is - it is not resistant to change.
- 33. Variance (σ2 ) The Variance is defined as: The average of the squared differences from the Mean. σ2 = Σ (Xi - x̄)2 / N-1 (if sample size ≤ 30) σ2 = Σ (Xi - x̄)2 / N
- 34. Standard deviation (σ) The Standard Deviation is a measure of how spread out numbers are. Its symbol is σ (the greek letter sigma) The formula is easy: it is the square root of the Variance. σ = √σ2
- 35. Coefficient of variance (Cv) The coefficient of variation represents the ratio of the standard deviation to the mean, and it is a useful statistic for comparing the degree of variation from one data series to another, even if the means are drastically different from each other Cv = Standard Deviation x 100 Mean