# Mean Deviation Calculator  Math Calculators
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## What is Mean Deviation?

In mathematics and statistics, deviation is a measure of difference between the observed value of a variable and some other value, often that variable's mean. The sign of the deviation reports the direction of that difference (the deviation is positive when the observed value exceeds the reference value). The magnitude of the value indicates the size of the difference.

Mean deviation is a statistical measure of the average deviation of values from the mean in a sample. It is calculated first by finding the average of the observations. The difference of each observation from the mean then is determined. The deviations then are averaged. This analysis is used to calculate how

Mean deviation is a statistical measure of the average deviation of values from the mean in a sample. It is calculated first by finding the average of the observations. The difference of each observation from the mean then is determined. The deviations then are averaged. This analysis is used to calculate how sporadic observations are from the mean.

In statistics, the absolute deviation is a measure of how much a particular sample deviates from the average sample. In simple terms, this means how much one number in a sample of numbers varies from the average of the numbers in the sample. Absolute deviation helps analyze data sets and can be a very useful statistic in a given report.

List data values in a column, for example:

2 5 7 10 12 14

Find the average of these values by adding them and then and dividing them by the number of values. In our example, the average is 8.3 (2+5+7+10+12+14=50, which is divided by 6).

Find the difference between each value and the average. Using our example, the differences are: 2 - 8.3 = 6.3 5 - 8.3 = 3.3 7 - 8.3 = 1.3 10 - 8.3 = 1.7 12 - 8.3 = 3.7 14 - 8.3 = 5.7

Calculate the average of the differences by adding them and dividing by the number of observations. The average of the differences in our example is 3.66: (6.3+3.3+1.3+1.7+3.7+5.7 divided by 6).

Find the average sample using one of three methods. The first method is by finding the mean. To find the mean, add together all of the samples and divide by the number of samples.

For example if your samples are 2, 2, 4, 5, 5, 5, 9, 10, 12, add them to get a total of 54. Then divide by the number of samples, 9, to calculate a mean of 6.

The second method of calculating the average is by using median. Arrange the samples in order from lowest to highest, and find the middle number. From the example, the median is 5.

The third method of calculating the average sample is by finding the mode. The mode is which ever sample occurs most. In the example, the sample 5 occurs three times, making it the mode.

Calculate the absolute deviation from the mean by taking the mean average, 6, and finding the difference between the mean average and the sample. This number is always stated as a positive number. For example, the first sample, 2, has an absolute deviation of 4, which is its difference from the mean average of 6. For the last sample, 12, the absolute deviation is 6.

Calculate the average absolute deviation by finding the absolute deviation of each sample and averaging them. From the example, calculate the absolute deviation from the mean for each sample. The mean is 6. In the same order, the absolute deviations of the samples is 4,4,2,1,1,1,3,4,6. Take the average of these numbers and calculate the average absolute deviation as 2.888 . This means the average sample is 2.888 from the mean.

### Standard Deviation

Standard deviation may be abbreviated SD, and is most commonly represented in mathematical texts and equations by the lower case Greek letter sigma σ, for the population standard deviation, or the Latin letter s, for the sample standard deviation. (For other uses of the symbol σ in science and mathematics see the main article.)

In addition to expressing the variability of a population, the standard deviation is commonly used to measure confidence in statistical conclusions. For example, the margin of error in polling data is determined by calculating the expected standard deviation in the results if the same poll were to be conducted multiple times. This derivation of a standard deviation is often called the "standard error" of the estimate or "standard error of the mean" when referring to a mean. It is computed as the standard deviation of all the means that would be computed from that population if an infinite number of samples were drawn and a mean for each sample were computed.

The standard deviation of a population and the standard error of a statistic derived from that population (such as the mean) are quite different but related (related by the inverse of the square root of the number of observations). The reported margin of error of a poll is computed from the standard error of the mean (or alternatively from the product of the standard deviation of the population and the inverse of the square root of the sample size, which is the same thing) and is typically about twice the standard deviation—the half-width of a 95 percent confidence interval.

In science, many researchers report the standard deviation of experimental data, and by convention, only effects more than two standard deviations away from a null expectation are considered statistically significant—normal random error or variation in the measurements is in this way distinguished from likely genuine effects or associations. The standard deviation is also important in finance, where the standard deviation on the rate of return on an investment is a measure of the volatility of the investment.

When only a sample of data from a population is available, the term standard deviation of the sample or sample standard deviation can refer to either the above-mentioned quantity as applied to those data, or to a modified quantity that is an unbiased estimate of the population standard deviation (the standard deviation of the entire population).

For example,

2, 4, 4, 4, 5, 5, 7, 9. Try It

These eight data points have the mean (average) of 5:

(2+4+4+4+5+5+7+9)/8 = 5

First, calculate the deviations of each data point from the mean, and square the result of each:

• (2-5)^{2}=(-3)^{2}=9

• (5-5)^{2}=0^{2}=0

• (4-5)^{2}=(-1)^{2}=1

• (5-5)^{2}=0^{2}=0

• (4-5)^{2}=(-1)^{2}=1

• (7-5)^{2}=2^{2}=4

• (4-5)^{2}=(-1)^{2}=1

• (9-5)^{2}=4^{2}=16

The variance is the mean of these values: ∑2 = (9+1+1+1+0+0+4+16)/8 = 4

and the standard deviation is equal to the square root of the variance:

∑ = √4 = 2