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Standard deviation

This document discusses standard deviation (SD), which is a measure of dispersion used commonly in statistical analysis. It describes how to calculate SD by finding the mean, deviations from the mean, sum of squared deviations, and variance. For large samples, the square root of the variance gives the SD. SD summarizes how much values vary from the mean, helps determine if differences are due to chance, and indicates appropriate sample sizes. For a normal distribution, about 68% of values fall within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD.

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M.Prasad Naidu MSc Medical Biochemistry, Ph.D,.
 It is an improvement over mean deviation  Measure of dispersion  Used most commonly in statistical analyses
Calculation of SD  First find the mean of series  Find the deviation or difference of individual measurement from mean  Next find the sum of squares of deviation or difference of individual measurements from their mean  Now find the variance (Var) – mean squared deviation var = ∑ (X - X) 2/ n
 If large sample size :  If sample less than 30 : Square root of variance that gives SD
For ex : Mean = 2+5+3+4+1/5 = 15/5 = 3 Var = 10/5 = 2
Uses of SD It summarizes the deviation of a large distribution from mean in one figure used as unit of variation  Indicates whether the variation of difference of an individual from mean is by chance  Helps in finding the SE which determines whether the difference between means of two similar samples is by chance or real  Helps in finding the suitable size of sample for valid conclusion.
 The shape of curve will depend upon mean and SD of which in turn depend upon the number and nature of observation.  In normal curve :-  Area b/w 1 SD on either side of mean will include approximately 68% of values in distribution  Area b/w 2 SD is 95%  Area b/w 3 SD is 99.7%  These limits on either side of mean are called “confidence limits” SD of normal curve
Standard normal curve  Smooth  bell shaped  Perfectly symmetrical  Based on infinity large number of observation  Total area of curve = 1  Mean = 0  SD = 1  Mean , median and mode all coincide
SD of normal curve Bell shaped curve will show an inflexion on the ascending as well as descending units of curve  If vertical lines are drawn from each of these points they will intersect the X axis on either side of the mean at an equal distance from it  A large portion of area under the normal curve has been included in portion of curve b/w the 2 points of inflexion
 The distance b/w the mean and point of inflexion either side is equal to SD and is denoted by a + sign prefixed to it to indicate that it extends on either side of mean.  If another vertical line is drawn an either side of mean at a distance equal to twice SD most of values in distribution table would have been included in this part of curve  In most cases, + SD will include 2/3 of sample values and mean + 2 SD will include 90% of values.
Standard deviation
Standard deviation