SSC CGL Tier 2 JSO / Paper 3 Statistics Study Material Day 6

SSC CGL Tier 2 JSO / Paper 3 Statistics Study Material Day 6

Standard Deviation

The standard deviation (often SD) is a measure of variability. When we calculate the standard deviation of a sample, we are using it as an estimate of the variability of the population from which the sample was drawn. For data with a normal distribution, about 95% of individuals will have values within 2 standard deviations of the mean, the other 5% being equally scattered above and below these limits. Contrary to popular misconception, the standard deviation is a valid measure of variability regardless of the distribution. About 95% of observations of any distribution usually fall within the 2 standard deviation limits, though those outside may all be at one end. We may choose a different summary statistic, however, when data have a skewed distribution.

The standard deviation is the most important and widely used measure of dispersion. This is also known as root mean square deviation, since it is the square root of the mean of the squared deviation of the sizes from arithmetic mean.

The standard deviation measures the absolute dispersion or variability of the distribution. The greater the amount of dispersion, the greater is the standard deviation.

Thus, standard deviation is defined as the positive square root of the squares of the deviation of all the sizes from their arithmetic mean.

Standard Deviation for Ungrouped Data

Standard Deviation for Frequency Distribution

Formula for computing standard deviation for a frequency distribution is

SSC CGL Tier 2; Paper 3 [Statistics] Study Material Day 6

Merits and Demerits of Standard Deviation

1. Its most important beauty is that it is free from the of Its most important beauty is that it is free from the compulsion of taking only absolute value in estimating mean deviation. So it is frequently applicable in different algebraic operations.
2. It took into account all individual observations and so any slight variation in any observations automatically got representation in standard deviation.
3. Through variance it easily reflects the aberration in data series.
4. It is the basis of relative measure of dispersion coefficient of variation (CV).
5. It is also an absolute measure of dispersion and so comparisons of data series in different units of comparisons of data series in different units of measurement are not tenable.
6. Its value changes if unit of measurement changed.
7. In a normal distribution, data are symmetrically distributed around mean(mean, median or mode all become identical) and mean σ covers 68.27 per cent of observations; mean 2σ covers 95 45 per cent of observations; mean 2σ covers 95.45 per cent of observations and mean 3σ covers 99.73 per cent of observations. This property is useful in dividing a data series into suitable groups or class.

Measures of Relative Dispersion

The relative dispersion of a data set, more commonly referred to as its coefficient of variation, is the ratio of its standard deviation to its arithmetic mean. In effect, it is a measurement of the degree by which an observed variable deviates from its average value. It is a useful measurement in applications such as comparing stocks and other investment vehicles because it is a way to determine the risk involved with the holdings in your portfolio.

Determine the arithmetic mean of your data set by adding all of the individual values of the set together and dividing by the total number of values.

Square the difference between each individual value in the data set and the arithmetic mean.

Add all of the squares calculated in Step 2 together.

Divide your result from Step 3 by the total number of values in your data set. You now have the variance of your data set.

Calculate the square root of the variance calculated in Step 4. You now have the standard deviation of your data set.

Divide the standard deviation calculated in Step 5 by the absolute value of the arithmetic mean calculated in Step 1.