Lecture 3 Measures of Central Tendency and Dispersion.pptx

Objectives:
Define measures of central tendency (mean, median, and mode)
Define measures of dispersion (variance and standard deviation).
Compute the measures of central tendency and Dispersion.
Learn the application of mean and standard deviation using Empirical rule and Tchebyshev’s theorem.
Measures of Central Tendency:
A measure of the central tendency is a value about which the observations tend to cluster.
In other words it is a value around which a data set is centered.
The three most common measures of central tendency are mean, median and mode.
A measure of the central tendency is a value about which the observations tend to cluster.
In other words it is a value around which a data set is centered.
The three most common measures of central tendency are mean, median and mode.
A measure of the central tendency is a value about which the observations tend to cluster.
In other words it is a value around which a data set is centered.
The three most common measures of central tendency are mean, median and mode.
A measure of the central tendency is a value about which the observations tend to cluster.
In other words it is a value around which a data set is centered.
The three most common measures of central tendency are mean, median and mode.
Why is it needed?
To summarize the data.
It provides with a typical value that gives the picture of the entire data set
Mean:
It is the arithmetic average of a set of numbers, It is the most common measure of central tendency.
Computed by summing all values in the data set and dividing the sum by the number of values in the data set Properties:
Applicable for interval and ratio data
Not applicable for nominal or ordinal data
Affected by each value in the data set, including extreme values.
Formula:
Mean is calculated by adding all values in the data set and dividing the sum by the number of values in the data set.
Median:
Mid-point or Middle value of the data when the measurements are arranged in ascending order.
A point that divides the data into two equal parts.
Computational Procedure:
Arrange the observations in an ascending order.
If there is an odd number of terms, the median is the middle value and If there is an even number of terms, the median is the average of the middle two terms.
Mode:
The mode is the observation that occurs most frequently in the data set.
There can be more than one mode for a data set OR there maybe no mode in a data set.
Is also applicable to the nominal data.
Comparison of Measures of Central Tendency in Positively Skewed Distributions:
Majority of the data values fall to the left of the mean and cluster at the lower end of the distribution: the tail is to the right Mean, median & mode are different When a distribution has a few extremely high scores, the mean will have a greater value than the median = positively skewed.
Majority of the data values fall to
the right of the mean and cluster at the upper end of the distribution= Negatively Skewed

1. Measures of Central Tendency and Dispersion Shakir Rahman BScN , MScN , MSc Applied Psychology, PhD Nursing (Candidate) University of Minnesota USA. Principal & Assistant Professor Ayub International College of Nursing & AHS Peshawar Visiting Faculty Swabi College of Nursing & Health Sciences Swabi Nowshera College of Nursing & Health Sciences Nowshera

2. Objective s Define measures of central tendency (mean, median, and mode) Define measures of dispersion (variance and standard deviation) . Compute the measures of central tendency and Dispersion. Learn the application of mean and standard deviation using Empirical rule and Tchebyshev’s theorem

3. Lets answer a few questions! What is the age of year 3 BScN students, class of 2020? How many hours per week, do BScN year 3 students spend on studying biostatistics? What is the height of boys enrolled in BScN program at AICNAHS ?

4. Measures of Central Tendency A measure of the central tendency is a value about which the observations tend to cluster. In other words it is a value around which a data set is centered. The three most common measures of central tendency are mean, median and mode. ( Munro ; Bluman, 2001 )

5. Why is it ne e ded? To summarize the data. It provides with a typical value that gives the picture of the entire data set ( Bluman, 2004)

6. M e a n It is the arithmetic average of a set of numbers, It is the most common measure of central tendency. Computed by summing all values in the data set and dividing the sum by the number of values in the data set Properties Applicable for interval and ratio data Not applicable for nominal or ordinal data Affected by each value in the data set, including extreme values .

7. Formula Mean is calculated by adding all values in the data set and dividing the sum by the number of values in the data set . X - = X 1 + X2 + X3 +. . . . . . Xn n ∑ = the sum of X = each individual value in the data set n= sample size (Sample) N = sample size (Population)

8. Sample Mean: Age of the patients coming to the clinic 57,86,42,38,90,66 X   X  X 1  X 2  X 3  . . .  X n n n  57  86  42  38  90  66 6  379 6  63.167

9. Pr ac ti c e time! The salaries of five f aculty members working at AICNAHS are : 20,000 , 18,000 24,000 30,000 22,000 Calculate the mean salary of the faculty members

10. X= 20,000+ 18,000+ 24,000+ 30,000+ 22,000 5 X = 114000 5 X= 22,800

11. Properties of mean Affected by the extreme high and low values in the data set 5, 6, 5, 8, 8, 7 5, 6, 5, 4, 20, 18 13,10, 11, 10, 0, 1 Therefore, it works best for symmetrical frequency distributions. (Bluman, 2004)

12. Median Mid-point or Middle value of the data when the measurements are arranged in ascending order. A point that divides the data into two equal parts Median: Computational Procedure Arrange the observations in an ascending order . If there is an odd number of terms, the median is the middle value and If there is an even number of terms, the median is the average of the middle two terms (Bluman, 2004)

13. Properties of Median Applicable for ordinal, interval, and ratio data Not applicable for nominal data Very simple and easy to calculate Unaffected by extremely large and extremely small values. It is used when one must determine whether the data values fall into the upper half or lower half of the distribution

14. Median Example (with an Odd Number) Arranged data in ascending order 14, 16, 21, 27, 27, 39, 45 Position of median = (n+1)/2 = (7+1)/2 = 4 There are 7 terms in the ordered array. The median is the 4th term, 27 years . If the 45 is replaced by 100, the median is 27 years . If the 14 is replaced by -103, the median is 27 year s .

15. Median Example(with an Even Number) Arranged data in ascending order 12, 14, 16, 21, 27, 27, 39, 45 There are 8 terms in the ordered array. Position of median = (n+1)/2 = (8+1)/2 = 4.5 The median is between the 4th and 5th terms, (21+27)/2 = 24 years If the 16 is replaced by 100, the median is 24years. If the 12 is replaced by -88, the median is 24years

16. Practice Following are the number of family members in selected houses in two communities. Calculate median: 17, 4, 15, 12, 18 Arrange in ascending order Since there are 5 values there fore, ( 5+1)= 3 2 Third number in the data set, after arrangement in ascending order, is the median

17. C o n td … 17, 4, 15, 12, 18, 11 Since there are 6 values there fore, (6+1)= 3.5 2 This means that the median is between the 3 rd and the 4 th term So mean of 12 and 15 will be the median for this data set

18. C o n td … Following are the values of pain intensity of 8 patients admitted with angina. These have been marked by the patients on a scale of 0-10. Calculate median: 9, 7, 4, 10, 3

19. Properties of Median 14, 15, 23, 28, 30 Not sensitive to extreme values; can be used for skewed data If 30 is replaced by 100, in the above example, still the median remains 23. If 14 is replaced by 2, in the above example, still the median remains 23.

20. Mod e The mode is the observation that occurs most frequently in the data set. There can be more than one mode for a data set OR there maybe no mode in a data set. Is also applicable to the nominal data

21. Practice Calculate mode for the following data set of : ▫ Sale of different brands of shampoo in a week ( P a n ten e= 1 , h e a d & shoulde r s= 2 , sun s i lk = 3) : 1, 1, 2, 3, 2, 3, 2, 1, 2, 2, 2, 1, 3 ▫ Pain scores of 5 patients 6, 4, 2, 10, 7 ▫ Scores of biostats exam 75, 80, 92, 42, 80, 68, 75

22. Mode Example: Nominal Data

23. F re qu e n c y Mode Example: Discrete variable 50 45 40 35 30 25 20 15 10 5 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 Number of golf clubs 13

24. Data Distribution Shapes

25. Comparison of Measures of Central Tendency in Normal Distribution In this, data values are evenly distributed on both sides of the mean Mean, median and mode are the same in symmetric shape 1

26. Comparison of Measures of Central Tendency in Bimodal Distribution Mean & median are the same Two modes different from mean and median 16 M ea n M edia n Mode M ode

27. Comparison of Measures of Central Tendency in Positively Skewe d 17 Distributions Majority of the data values fall to the left of the mean and cluster at the lower end of the distribution: the tail is to the right Mean, median & mode are different When a distribution has a few extremely high scores, the mean will have a greater value than the median = positively skewed Mean > Median > Mode

28. Comparison of Measures of Central Tendency in Negatively Skewed Distributions Majority of the data values fall to the right of the mean and cluster at the upper end of the distribution: the tail is to the left Mean, median & mode are different When a distribution has a few extremely low scores, the mean will have a lower value than the median = negatively skewed Mode > Median > Mean

29. Mean M ed i a n No Mode

30. No (so Lev inal, Ordinal, and metimes) Interval/Ratio- el Data Interval/Ratio-Level Data Comparison of Measures of Central T e nd e n c y Mode Most frequently occurring value Nominal , Ordinal, and (sometimes) Interval/Ratio- Level Data Ordinal-Level Data and Interval/Ratio-Level data (particularly when skewed) M e dian Exact center (when odd N) of rank-ordered data or average of two middle values . Mean Arithmetic average (Sum of Xs / n )

31. Use of measures of central tendency Mean is the most common measure which uses each value in the data set. It is best to be used when the distribution is symmetrical If the distribution is skewed, then median is a better measure of central tendency as it is unaffected by the extreme values Mode depicts the most preferred or the most popular product, candidate etc. Can be calculated for all levels of data but is not very meaningful for interval and ratio scale data

32. Recap

33. Measures of Dispersion

34. Measures of Dispersion Calculate mean for the following data sets: 5, 6, 8, 10, 12, 14, 15 1, 4, 8, 10, 12, 16, 19 Then what’s the difference between the following data sets ? The difference is in the spread of both data sets; the spread of the second data set is more than the first one (Bluman, 2004)

35. Why is it important to know the dispersion of data? Example: Scores of student A: 23, 32, 74, 56, 48 Scores of student B: 55, 67, 63, 57, 65 Ages of sample 1: 12, 15, 22, 34, 50, 56 Ages of sample 2: 10, 12, 14, 18, 17, 16 It shows the consistency and homogeneity/heterogenity in the data

36. Measures of Variability Measures of variability describe the spread or the dispersion of a set of data. If all the values in a data set are same there is no dispersion BUT dispersion is present when values are not same in data set. The amount of dispersion may be small, when the values though different are close together Common Measures of Variability: V aria n c e Stan d ard Deviation Range Coefficient of Variation

37. Range The difference between the largest and the smallest values in a set of data Simple to compute Ignores all data points except the two extremes. Example: Range = Largest – Smallest = 48 - 35 = 13 The range is quick to compute but fails to be very useful since it considers only the extreme values and does not take into consideration the bulk of the observations. It is not widely used.

38. Variance Variance is the preferred measure of variation for most statistical analys i s Uses all the values in data and defined in terms of the deviation of values from their mean: If the values of data lie close to their mean, the dispersion is less than when they are scattered over a wide range Population Variance : Sample Variance :

39. Computation S 2 = ∑ ( X- X) 2 n -1 ∑= the sum of X= each individual value in the data set X= Sample mean n= sample size

40. Example Calculate variance for the following sample of weight losses (in Kgs) by 5 people 0, 15, 10, 22, 3 S 2 = 318 = 79. 5 Kg 4 X X- X (X- X) 2 15 10 22 03 0-10= -10 15-10= 5 10-10= 22-10= 12 3-10 = -7 100 25 144 49 318

41. Sample Variance Average Systolic Blood Pressure of the values from the 6 Cardio patients are: 130, 138, 188, 188, 112, 162, and 160. A rithmetic mean (=148.3 mmHg)

42. Standard Deviation Standard Deviation is defined as the square root of the variance It is more convenient to express the variation in the original units by taking the square root of the variance Population Standard Deviation Sample Standard Deviation:

43. Standard D e v i a t i o n It is the square root of variance Population standard deviation is denoted by σ Sample standard deviation is denoted by s S= √S 2 (Bluman, 2004)

44. Sample Standard deviation - Average distance of the values from the arithmetic mean - Square root of the sample variance Where Mean is 27years S = 794 = 132.33 6 S 2 = √S = √132.33yrs S = 11.5 yrs

45. Mean & Standard deviation for Grouped data When data is presented in grouped form, the mean and variance are computed by following equations: Where “fi” is the number of observations in the respective class interval and c is the number of classes.

46. The Variance and Standard Deviation When the data are clustered about the mean, the variance and standard deviation will be somewhat small.

47. The Variance and Standard Deviation When the data are widely scattered about the mean, the variance and standard deviation will be somewhat large. 28

48. Coefficient of Variation When two data sets have the same unit, their standard deviations can be compared directly. For e.g. we can compare the standard deviations of the mileage of two brands of cars. If in a particular year, the standard deviation of the mileage of Mehran is 360 miles and of Vitz is 200 miles, then we can say that there is more variation in the mileage of mehran than Vitz. It was possible to compare the two SDs because their units were same

49. C o n t d . . . But if we want to compare the SD of two variables whose units are different then what? For instance, a manager wants to compare the SD of number of sales done by salesmen per year to the SD of commission made by these sales persons In such cases Coefficient of Variation is calculated (Bluman, 2004)

50. Coefficient of Variation One important application of the mean and the standard deviation is the coefficient of variation. CV ar = Standard Deviation x 100 Mean The coefficient of variation depicts the size of the standard deviation relative to its mean. Since both stan d ard d e v i a t ion and t h e m e an rep r ese n t t h e sa m e u n its the units cancel out and the coefficient of variation becomes a pure number.

51. Coefficient of Variation The CV is useful for comparing scatter of variables measured in different units Examples: The mean number of parking tickets issued in a neighborhood over a four-month period was 90, and the standard deviation was 5. The average revenue generated from the tickets was $5,400, and the standard deviation was $775. Compare the variations of the two variables.

52. Coefficient of Variation Solution: 34

53. Coefficient of Variation - Explanation of the term – population coefficient of variation: the population coefficient of variation is defined as the population standard deviation divided by the population mean of the data set. NOTE: The population CV ar has the same properties as the sample CV ar .

54. R e f ere n c es Bluman, A. (2004). Elementary statistics: A step by step approach. Boston: Mc Graw Hill.

55. Acknowledgments Dr Tazeen Saeed Ali RM, RM, BScN, MSc ( Epidemiology & Biostatistics), Phd (Medical Sciences), Post Doctorate (Health Policy & Planning) Associate Dean School of Nursing & Midwifery The Aga Khan University Karachi. Kiran Ramzan Ali Lalani BScN, MSc Epidemiology & Biostatistics Aga Khan University Karachi