What Is Standard Deviation
Measures of dispersion is one of the important sections of statistics. It helps in finding variability of data in a distribution or series. The amount and degree of variation are two factors that helps in measuring dispersion. The use of amount of variation in measuring dispersion is known as absolute measure of dispersion. Standard deviation is a positional measure, which is a subsection of absolute measure. Standard deviation is that positional measure which works on all the items or variables in a distribution. Just like Mean deviation, standard deviation is based on all the listed item of a series. Standard deviation is used for calculating population or change in population of a region.
Understanding standard deviation
The definition and calculation of standard deviation varies with the type of data, ungrouped or grouped and the use of mean. Standard deviation in its simplest form is nothing but a refined version of mean deviation. It neutralizes all the flaws of mean deviation.
In standard deviation, the deviations counted by subtracting every item with the median in mean deviation is squared. This squared deviations are summed and divided by the number of items in the list. The square root of the result is the standard deviation. This forms the basic formula of standard deviation.
The formula: √ ( ( ∑ d²) /n)
Grouped data is the data, which is represented in the form of class interval, like 10-20, and you will need to find the midpoint in order to work with it. Whereas, ungrouped data is never represented in the form of class intervals. There are many formulas of standard deviation and its use varies with type of data given( ungrouped or grouped). Here are the formulas
Formula for ungrouped data
Using mean: √ ( ( ∑ d²) /n)
Here, ∑ d² is the square of the deviations found by subtracting each item with the mean of the distribution and n stands for the number of items.
Without using mean, Direct method: √ ( ( ∑ x²) /n) – ( ∑ x) /n)²) here ∑ x² and ∑ x are the sum of the squares of the items and sum of the items respectively.
Deviation Method: In this formula a standard point of reference is taken, say R and the value of deviation, say d is found by subtracting every item, say x from R. Here, the sum of squares of deviation is denoted by n. So, d = R-x, and the formula is √ ( ( ∑ d²) /n) – ( ∑ d) /n)²)
Formula for grouped data
Here, f stands for given frequency.
Using mean: √ ( ( ∑ fd²) / ∑ f). Mid points for each class interval is calculated and its mean is found. Thus, d is found by subtracting each midpoints with the mean.
Without using mean, Direct method : √ ( ( ∑ fx²) /∑ f) – ( ∑ fx) /∑ f)²). Here x is the midpoint of a class interval.
Deviation method: √ ( ( ∑ fd²) /∑ f) – ( ∑ fd) /∑ f)²) x c
The above formula is used when the class intervals are of same size. The midpoints of the class intervals are denoted as x, and we assume a mean from the list if midpoints, usually the number lying in the middle of the distribution. The difference between every midpoint is always same and that number is assigned as c. In order to calculate d, we subtract x from A and divide the result with c. So, d= (x-A)/c
Importance of standard deviation
Both quartile deviation and mean deviation have some flaws, which cannot be overlooked. Quartile deviation doesn’t work in situations such as unstable range. Whereas, if you are using mean deviation then you will not be able to work on different algebraic signs of deviations as mean deviation fails to work on algebraic signs of deviations. Standard deviation is the only measure of dispersion which overcomes such flaws.