Biosttistics for ayurveda students and yoga students

biostatisticss

1. Measures of dispersion Measures of variability of observations help to find how individual observations are dispersed around the mean of a large series. They may also be called, measures of dispersion, variation or scatter as against the averages which are measures of central tendency. Measures of variability of individual observations: Range Interquartile range Mean deviation Standard deviation Coefficient of variation.

2. Range Variability being a biological characteristic, no single measurement or observation of a variable is considered as an indicator of normality. A range defines the normal limits of a biological characteristic. Some ranges are given below:

3. Range for a biological character such as height, hemoglobin, etc. is worked out after measuring the characteristic in large number of healthy persons of the same age, sex, class, etc. It is calculated by finding the difference between the maximum and minimum measurements in the series. Often the two measurements rather than their range are given, e.g. in the series of whole size 3, 1, 6, 10 and 9 mmHg, the range is 10–1 or 10 to 1 and is given as (1–10) in most of the published works.

4. Range is the simplest measure of dispersion and is usually employed as a measure of variability in medical practice by one who has little knowledge of statistical methods. It indicates the distance between the lowest and the highest. Though of interest, it is not a satisfactory measure as it is based only on two extreme values, ignoring the distribution of all other observations within the extremes. Ordinarily in medical practice, the normal range covers the observations falling in 95% confidence limits

5. Interquartile Range Definition The interquartile range defines the difference between the third and the first quartile. Quartiles are the partitioned values that divide the whole series into 4 equal parts. So, there are 3 quartiles. First Quartile is denoted by Q 1  known as the lower quartile, the second Quartile is denoted by Q 2  and the third Quartile is denoted by Q 3  known as the upper quartile. Therefore, the interquartile range is equal to the upper quartile minus lower quartile. Interquartile Range Formula Interquartile range = Upper Quartile – Lower Quartile = Q ­3  – Q ­1

6. Example : Determine the interquartile range value for the first ten prime numbers. Solution: Given: The first ten prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 This is already in increasing order. Here the number of values = 10 10 is an even number. Therefore, the median is mean of 11 and 13 That is Q 2  = (11 + 13)/2 = 24/2 = 12. Now we have to get two parts i.e. lower half to find Q 1  and the upper half to find Q 3 . Q 1  part :  2, 3, 5,7,11 Here the number of values = 5 5 is an odd number. Therefore, the center value is 5, that is Q 1 = 5 Q 3  part :  13, 17, 19, 23, 29 Here the number of values = 5 5 is an odd number. Therefore, the center value is 19, that is Q 3 = 19 The subtraction of Q1 and Q 3  value is 19 – 5 = 14 Therefore,  14 is the interquartile range value.

7. Advantages The important of interquartile range is that it can be used as a measure of variability if the extreme values are not being recorded exactly (as in case of open-ended class intervals in the frequency distribution). Other advantageous feature is that it is not affected by extreme values. The interquartile range is the best measure of variability for skewed distributions or data sets with outliers. Because it's based on values that come from the middle half of the distribution, it's unlikely to be influenced by outliers.

8. Mean Deviation Definition The mean deviation is defined as a statistical measure that is used to calculate the average deviation from the mean value of the given data set. The mean deviation of the data values can be easily calculated using the below procedure.  Step 1: Find the mean value for the given data values Step 2: Now, subtract the mean value from each of the data values given (Note: Ignore the minus symbol) Step 3: Now, find the mean of those values obtained in step 2.

9. Mean Deviation Formula The formula to calculate the mean deviation for the given data set is given below. Mean Deviation = [Σ |X – µ|]/N Here,  Σ represents the addition of values X represents each value in the data set µ represents the mean of the data set N represents the number of data values | | represents the absolute value, which ignores the “-” symbol

10. Example : Determine the mean deviation for the data values 5, 3,7, 8, 4, 9. Given data values are 5, 3, 7, 8, 4, 9. We know that the procedure to calculate the mean deviation. First, find the mean for the given data: Mean, µ = ( 5+3+7+8+4+9)/6 µ = 36/6 µ = 6 Therefore, the mean value is 6. Now, subtract each mean from the data value, and ignore the minus symbol if any (Ignore”-”) 5 – 6 = 1 3 – 6 = 3 7 – 6 = 1 8 – 6 = 2 4 – 6 = 2 9 – 6 = 3 Now, the obtained data set is 1, 3, 1, 2, 2, 3. Finally, find the mean value for the obtained data set Therefore, the mean deviation is  = (1+3 + 1+ 2+ 2+3) /6 = 12/6 = 2 Hence, the mean deviation for 5, 3,7, 8, 4, 9 is 2.

11. Advantages It is based on all the data values provided, and hence it will give a better measure of dispersion. It is easy to understand and calculate. Mean deviation is used to compute how far the values in a data set are from the center point. Mean Deviation tells us how far, on average, all values are from the middle. 

12. Standard Deviation (SD) Standard deviation is an improvement over mean deviation as a measure of dispersion and is used most commonly in statistical analyses. It is computed by following six steps: a. Calculate the mean. b. Find the difference of each observations from the mean. c. Square the differences of observations from the mean. d. Add the squared values to get the sum of squares of the deviation. e. Divide this sum by the number of observations minus one to get mean-squared deviation, called Variance (σ2 ). f. Find the square root of this variance to get root-mean squared deviation, called standard deviation. Having squared the original, reverse the step of taking square root.

13. Uses of standard deviation It summarizes the deviations of a large distribution from mean in one figure used as a unit of variation. Indicates whether the variation of difference of an individual from the mean is by chance, i.e. natural or real due to some special reasons. Helps in finding the standard error which determines whether the difference between means of two similar samples is by chance or real. It also helps in finding the suitable size of sample for valid conclusions

14. Coefficient of variation (CV) It is used to compare the variability of one character in two different groups having different magnitude of values or two characters in the same group by expressing in percentage. The coefficient of variation is calculated from standard deviation and mean of the characteristic. The ratio of SD and mean is found in percentage. Thus SD expressed as percentage of mean is the coefficient of variation.