** Deviation vs Standard Deviation
**

** Deviation vs Standard Deviation
**

In descriptive and inferential statistics, several indices are used to describe a data set corresponding to its central tendency, dispersion and skewness. In statistical inference, these are commonly known as estimators since they estimate the population parameter values.

Dispersion is the measure of the spread of data around the center of the data set. Standard deviation is one of the most commonly used measures of dispersion. The deviations of each data point from the mean is taken into account when calculating the standard deviation. Hence, one can argue that the standard deviation along with the mean will provide an almost sufficient picture about a data set.

Consider the following data set. The weights of 10 people (in kilograms) are measured to be 70, 62, 65, 72, 80, 70, 63, 72, 77 and 79. Then the mean weight of the ten people (in kilograms) is 71 (in kilograms).

**What is deviation?**

In statistics, deviation means the amount by which a single data point differs from a fixed value such as the mean. In general, let k be a fixed value and x_{1},x_{2}, …, x_{n} denote a data set. Then, the deviation of x_{j }from k is defined to be (x_{j}– k).

For example, in the above data set the respective deviations from the mean are (70 – 71) = -1, (62 – 71) = -9, (65 – 71) = -6, (72 – 71) = 1, (80 – 71) = 9, (70 – 71) = -1, (63 – 71) = -8, (72 – 71) = 1, (77 – 71) = 6 and (79 – 71) = 8.

**What is standard deviation?**

When data from the whole population can be taken into account (for example in the case of a census), it is possible to calculate the population standard deviation. To calculate the standard deviation of the population, first the deviations of data values from the population mean are calculated. The root mean square (quadratic mean) of deviations is called the population standard deviation. In symbols, σ = √{ ∑(x_{i}-µ)^{2} / n} where µ is the population mean and n is the population size.

When data from a sample (of size n) is used to estimate parameters of the population, the sample standard deviation is calculated. First the deviations of data values from the sample mean are calculated. Since the sample mean is used in place of the population mean (which is unknown), taking the quadratic mean is not appropriate. In order to compensate for the use of the sample mean, the sum of squares of deviations is divided by (n-1) instead of n. The sample standard deviation is the square root of this. In mathematical symbols, S = √{ ∑(x_{i}-ẍ)^{2} / (n-1)}, where S is the sample standard deviation, ẍ is the sample mean and xi’s are the data points.

In the previous data set, the sum of squares of deviation is (-1)^{2} + (-9)^{2} + (-6)^{2} + 1^{2} + 9^{2} + (-1)^{2} + (-8)^{2} + 1^{2} + 6^{2} + 8^{2} = 366. Thus, the population standard deviation is √(366/10) = 6.05 (in kilograms). (Assuming that the population under consideration is comprised of the 10 people from whom the data was taken).

• Standard deviation is a statistical index and an estimator, but deviation is not. • Standard deviation is a measure of dispersion of a cluster of data from the center, whereas deviation refers to the amount by which a single data point differs from a fixed value. |

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