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Mean, Variance and Standard Deviation | Engineering Mathematics - Civil Engineering (CE) PDF Download
Mean
Mean is average of a given set of data. Let us consider below example
2, 4, 4, 4, 5, 5, 7, 9
These eight data points have the mean (average) of 5:
Where μ is mean and x1, x2, x3…., xi are elements.Also note that mean is sometimes denoted by x
Variance
Variance is the sum of squares of differences between all numbers and means.
Deviation for above example. 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 = 02 = 0
(4 - 5)2 = (-1)2 = 9 (5 - 5)2 = 02 = 0
(4 - 5)2 = (-1)2 = 9 (7 - 5)2 = 22 =4
(4 - 5)2 = (-1)2 = 9 (9 - 5)2 = 42 = 16.
variance =
Where μ is Mean, N is the total number of elements or frequency of distribution.
Standard Deviation
Standard Deviation is square root of variance. It is a measure of the extent to which data varies from the mean.
Standard Deviation (for above data) = √4 = 2
Why did mathematicians chose a square and then square root to find deviation, why not simply take the difference of values?
One reason is the sum of differences becomes 0 according to the definition of mean. Sum of absolute differences could be an option, but with absolute differences, it was difficult to prove many nice theorems.
Some Interesting Facts
- Value of standard deviation is 0 if all entries in input are same.
- If we add (or subtract) a number say 7 to all values in the input set, mean is increased (or decreased) by 7, but standard deviation doesn’t change.
- If we multiply all values in the input set by a number 7, both mean and standard deviation is multiplied by 7. But if we multiply all input values with a negative number say -7, mean is multiplied by -7, but the standard deviation is multiplied by 7.
- Standard deviation and varience is a measure which tells how spread out numbers is. While variance gives you a rough idea of spread, the standard deviation is more concrete, giving you exact distances from the mean.
- Mean, median and mode are the measure of central tendency of data (either grouped or ungrouped).
The document Mean, Variance and Standard Deviation | Engineering Mathematics - Civil Engineering (CE) is a part of the Civil Engineering (CE) Course Engineering Mathematics.
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FAQs on Mean, Variance and Standard Deviation - Engineering Mathematics - Civil Engineering (CE)
| 1. What is the formula to calculate the mean? |  |
Ans. The formula to calculate the mean is to sum up all the values in a dataset and then divide it by the total number of values in the dataset.
| 2. How is the variance calculated? | |
Ans. Variance is calculated by taking the difference of each value in a dataset from the mean, squaring the differences, summing up all the squared differences, and then dividing it by the total number of values in the dataset.
| 3. What does the standard deviation represent? | |
Ans. The standard deviation represents the average amount of deviation or dispersion of data points from the mean in a dataset. It measures how spread out the values are from the average.
| 4. Can standard deviation be negative? | |
Ans. No, standard deviation cannot be negative. It is always a non-negative value since it is the square root of the variance, which is always non-negative.
| 5. How can mean, variance, and standard deviation be useful in data analysis? | |
Ans. Mean, variance, and standard deviation are useful statistical measures in data analysis. The mean helps to understand the central tendency of the data, while the variance and standard deviation provide insights into the spread or dispersion of the data points. These measures can help in comparing and summarizing datasets, identifying outliers, and making statistical inferences.
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