Standard deviation

Range of grouped data

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

Mean deviation

Every value of the variable differs from the sample mean by some specific amount which is called its 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.

Mean deviation of grouped data

M.D. = ∑ IfdI/ ∑f

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. i. 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.

How to compute standard deviation

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. ** go back to slide for formula and in depth explination

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. ** go back to slides (29-30)

Standard deviation

It indicates the difference between a group of values and their mean, taking all of the data into account. The standard deviation plays an important role in many tests of statistical significance. The larger the standard deviation, the more the values differ from the mean, and therefore the more widely they are spread out.

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

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