Measures of dispersion

Deciles

9 in number divide the distribution into 10 equal parts, first decile or 10th percentile will divide the distribution into 10% and 90% while 9th decile will divide into 90% and 10% and 5th decile will be same as median. So, median of a variable can also be called as second quartile Q2, 5th decile D5 or 50th percentile P50.

Computation of mean deviation of ungrouped data

Calculate the mean from the data Calculate the deviation from the mean. Sum up all deviations. All deviations are treated as positive. Divide sum of all deviations by the total number of observations. E.g 15, 17, 19, 25, 30, 35, 48 Mean of this data is = ∑X/N = 189/7 = 27

Percentiles

Centiles or percentiles are values in a series of observations arranged in ascending order of magnitude which divide the distribution into 100 equal parts. Thus, the median is 50th centile. The 50th percentile will have 50% observations on either side. Accordingly, 10th percentile should have 10% observations to the left and 90% to the right. If children at age 3½ years form 10th percentile, it means 10% of entire population is below 3½ years of age and 90% is above that age. Thus, percentiles are used to divide a distribution into convenient groups.

Calculation of standard deviationfor ungrouped series

First find the mean of series. It is calculated as the sum of observations divided by their number Then find the deviations or differences of the individual measurements from the mean. Next find the sum of the squares of deviations or differences of individual measurements from their mean. This can be expressed by the formula, Now find the variance (σ2) which is mean squared deviation, i.e. sum of squares of deviation, divided by the number of independent observations. This number is not total but one less than the total number or measurements or observations, (n) in the series, therefore, divide by n - 1. It is also called the degrees of freedom in statistical terms and gives an unbiased estimate of variance. Hence, variance Lastly, determine square root of the variance. That gives the standard deviation which in fact is the square root of the mean squared deviation.

Quartiles calculation

First quartile ( Q1) or the 25th percentile Second quartile (Q2) also called the median or the 50th percentile Third quartile (Q3) or the 75th percentile Interquartile range (IQR) is the difference between the upper and lower quartiles. (IQR = Q3 - Q1)

Range of grouped data

Formula = upper true limit of the highest class - lower true limit of the lowest class

Uses of standard deviation

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

Standard deviation

It indicates the difference between a group of values and their mean, taking all of the data into account. The larger the standard deviation, the more the values differ from the mean, and therefore the more widely they are spread out.

Quintiles

Quintiles, 4 in number divide the distribution into 5 equal parts. So, 20th percentile or first quintile will have 20% observations falling to its left and 80% to its right.

Coefficient of variation

It is a measure used to compare relative variability. It compares the variability irrespective of the units of measurement used in two or more distributions. Coefficient of variation (CV) 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.

Mean deviation

Mean deviation is an average mean of the deviations of values from central value or central tendency. Thus, the mean deviation can be defined 'as the mean of all the deviations in a given set of data obtained from an average' If the deviation is greater than the mean, the deviation is positive, but if it is less than the mean, the deviation is negative.

Range

The range of distribution is the difference between the largest and the smallest values in a set of observations. Range = highest value in the series of data - lowest value of that series

Quartiles

They are 3 different points located on the entire range of a variable such as height—Q1, Q2 and Q3. Q1 or lower quartile will have 25% observations of heights falling on its left and 75% on its right; Q2 or median quartile will have 50% observations on either side and Q3 or upper quartile will have 75% observations on its left and 25% on its right.

Mean deviation of grouped data

Where, ∑ I fd I = sum of multiplication of each frequency and deviation from mean ∑f = sum of all frequencies.

Calculation of standard deviation for grouped series

Without finding the mean, SD can be calculated in grouped series also by almost the same steps as for mean. 1. Make the frequency table. ii. Place 0 opposite the middle group (working mean). iii. Reduce the values to working units as in the case of mean dividing the differences from working mean by class interval. iv. Just as frequency is multiplied by working units for finding mean, similarly multiplied frequency by squares of working units for finding the standard deviation. v. Then apply the formula and find the mean and SD in working units as before. vi. Convert this value of SD in real units. Multiply SD in working units by the size of class interval. Here variance is calculated by some other formula. Variance = Where ∑fX² = sum of squared deviation and (∑fX)² = sum of deviation from assumed mean

Calculation of Percentiles

arrange the values in ascending order Formula (Nearest rank method) P = ( k ÷ 100) × (n) kth = The percentile one wishes to calculate. P= Represents the position within the distribution that marks the percentile one wishes to calculate. n= The total number of values in the distribution