5th lecture on Measures of dispersion for

5th lecture on Measures of dispersion for Allied Health professional students

1. Measures of dispersion/ Variation Mr. Bashir Ullah Demonstrator Cardiac Perfusion Technology. MS Epi & Biostatistics KMU,IPH&SS

2. Objectives By the end of this lecture, we should be able to : Compute & list uses for measures of dispersion: range, variance, standard deviation. Compute mean & standard deviation for grouped data.

3. MEASURES OF DISPERSION Measures of central tendency (mean, median, mode, GM and HM) do not provide all information about the observations contained in a data set that how the individual observations are scattered around the central value. It is possible with the help of measures of dispersion. A single value which measure that how the individual observations of a data set are scattered/dispersed around the central value, is called measure of dispersion. A measure of dispersion can be expressed in an absolute value or relative value.

4. COMMONLY USED MEASURES OF DISPRSION

5. 1.Range The range is the simplest of the three measures. The range is the highest value minus the lowest value of a data set . The symbol R is used for the range . Mathematically, range is defined as: Range = X m – X

6. Example The following data indicate the amount of fill (in ml) of 5 different bottle by a soft drink company. The data is 12.5, 12.3, 12, 13, 12.8 So, Range = 13-12 = 1 ml . Assessment Question Find Range of the following data: 13,23,11,25,18,20,40

7. Range For Group data For grouped data, the range is difference between the upper class boundary of the highest class and the lower class boundary of the lowest class. Example: Find Range of the following data. Groups 60-64 65-69 70-74 75-79 80-84 85-89 F 127 133 142 178 140 130 Solution: Groups Class Boundary 60-64 65-69 70-74 75-79 80-84 85-89 59.5-64.5 64.5-69.5 69.5-74.5 74.5-79.5 79.5-84.5 84.5-89.5 Xm= 89.5, Xo= 59.5 Range= Xm – X Range= 89.5 – 59.5= 30

8. CO-EFFICIENT OF RANGE The range is absolute measure of dispersion, its relative measure is known as co-efficient of range. co-efficient of range= Example: The IQ’s of 5 members of a family are 108,112,127,118 and 113. find the Co-efficient of Range.  

9. Advantages and Disadvantages of Range Advantages: It is easy to understand and explain. It is easy to calculate. It is used in statistical quality control. It is useful as a rough measure of variation. Disadvantages: It is a poor measure of variability as it accounts only two extreme value of data. It is not possible of compute range in the case of open-end distribution. It is not a satisfactory measure in the case of frequency distribution.

10. 2.VARIANCE The mean of the squared deviation of all the observation from their mean, is known as variance “OR” The variance is the average of the squares of the distance each value is from the mean. Mathematically, variance is defined as:

11. VARIANCE FOR FREQUENCY DISTRIBUTION For frequency distribution (discrete or continuous), variance is expressed as. It is important to mention that variance is always be positive number (or non-negative). It can not be a negative value. S quaring the standard deviation gives the variance.

12. 3.Standard Deviation The Standard Deviation is a measure, which describes how much individual measurements differ, on the average, from the mean A large standard deviation shows that there is a wide scatter of measured values around the mean. Small standard deviation shows that the individual values are concentrated around the mean with little variation among them . Standard deviation: it is the square root of the variance

13. STANDARD DEVIATION Standard deviation is the square root of variance and is expressed as, mathematically.

14. STANDARD DEVIATION FOR FREQUENCY DISTRIBUTION For frequency distribution (discrete or continuous), standard deviation is expressed as: Note: Standard deviation will always be a positive number (or non-negative). It can not be a negative value. Standard deviation is an absolute measure of dispersion.

15. FINDING SAMPLE/POPULATION VARIANCE AND S.D Step 1: Find the mean for the data. (Population Data) (sample data) Step 2: Subtract the mean from each data value. Step 3: Square each result . Step 4: Find the sum of the squares. Step 5: Divide the sum by N to get the variance. Step 6: Take the square root of the variance to get the standard deviation.  

16. Example Data: 1,2,3,4,5 x i x X i - x (X i – x) 2 S (X i – x) 2 /n 1 3 1-3 (-2) 2 =4 2 2-3 (-1) 2 =1 3 3-3 (0) 2 =0 4 4-3 (1) 2 =1 5 5-3 (2) 2 =4 15 15/5=3 10 S 2 = 10/5=2 S = √2 = 1.41 Step:1 Step:2 Step:3 Step:4 Step:5 Step:6

17. Advantages and disadvantages of standard deviation It is based on all the values . Much use in inferential statistics ,play role in normal distribution. It is less affected by fluctuations of sampling. It is useful comparing number of different sets of data. Disadvantages It is difficult to calculate. It is affected by extreme value. It gives more weight to extreme values and less of those which are near the mean.

18. example Let us say that a group of patients enrolling for a trial had a normal distribution for weight. The mean weight of the patients was 80 kg. For this group, the SD was calculated to be 5 kg. 1 SD below the average is 80 – 5 = 75 kg. 1 SD above the average is 80 + 5 = 85 kg . ± 1 SD will include 68.2% of the subjects, so 68.2% of patients will weigh between 75 and 85 kg. 95.4 % will weigh between 70 and 90 kg (±2 SD ). 99.7 % of patients will weigh between 65 and 95 kg (±3 SD).

19. C v C. V = S.D/Mean it is unitless. When we want to compare two or more data sets, the co-efficient of variation is used. Use the coefficient of variation when you want to compare variability between: Groups that have means of very different magnitudes. Characteristics that use different units of measurements. It is also known as the relative standard deviation (RSD ).