Variance is the expected value of the squared variation of a random variable from its mean value, in probability and statistics. Informally, variance estimates how far a set of numbers (random) are spread out from their mean value. Show
The value of variance is equal to the square of standard deviation, which is another central tool. Variance is symbolically represented by σ2, s2, or Var(X). The formula for variance is given by: \(\begin{array}{l}Var (X) = E[( X – \mu)^{2}]\end{array} \) Variance is a measure of how data points differ from the mean. According to Layman, a variance is a measure of how far a set of data (numbers) are spread out from their mean (average) value. Variance means to find the expected difference of deviation from actual value. Therefore, variance depends on the standard deviation of the given data set. The more the value of variance, the data is more scattered from its mean and if the value of variance is low or minimum, then it is less scattered from mean. Therefore, it is called a measure of spread of data from mean. For the purpose of solving questions, the formula for variance is given by: \(\begin{array}{l}Var (X) = E[( X – \mu)^{2}]\end{array} \) Put into words; this means that variance is the expectation of the squared deviation of a random set of data from its mean value. Here, X = Random variable “µ” is equal to E(X) so the above equation may also be expressed as, Var(X) = E[(X – E(X))2] Var(X) = E[ X2 -2X E(X) +(E(X))2] Var(X) = E(X2) -2 E(X) E(X) + (E(X))2 Var(X) = E(X2) – (E(X))2 Sometimes the covariance of the random variable itself is treated as the variance of that variable. Symbolically, Var(X) = Cov(X, X) FormulaAs we know already, the variance is the square of standard deviation, i.e., Variance = (Standard deviation)2= σ2 The corresponding formulas are hence, \(\begin{array}{l}Population\ standard\ deviation\ \sigma = \sqrt{\frac{\sum (X-\mu )^{2}}{N}}\end{array} \) \(\begin{array}{l}Sample\ standard\ deviation\ s = \sqrt{\frac{\sum (x-\overline{x})^{2}}{n-1}}\end{array} \) Where X (or x) = Value of Observations μ = Population mean of all Values n = Number of observations in the sample set \(\begin{array}{l}\bar{x}= Sample\ mean\end{array} \) N = Total
number of values in the population PropertiesThe variance, var(X) of a random variable X has the following properties.
Var(X1 + X2 +……+ Xn) = Var(X1) + Var(X2) +……..+Var(Xn). Variance and Standard DeviationStandard deviation is the positive square root of the variance. The symbols σ and s are used correspondingly to represent population and sample standard deviations. Standard Deviation is a measure of how spread out the data is. Its formula is simple; it is the square root of the variance for that data set. It’s represented by the Greek symbol sigma (σ). How to Calculate VarianceVariance can be calculated easily by following the steps given below:
Say if x1, x2, x3, x4, …,xn are the given values. Therefore, the mean of all these values is: x̄ = (x1+x2+x3+…+xn)/n Now subtract the mean value from each value of the given data set and square them. (x1-x̄)2, (x2-x̄)2, (x3-x̄)2,…….,(xn-x̄)2 Find the average of the above values to get the variance. Var (X) = [(x1-x̄)2+ (x2-x̄)2+ (x3-x̄)2+…….+(xn-x̄)2]/n Hence, the variance is calculated. Example of VarianceLet’s say the heights (in mm) are 610, 450, 160, 420, 310. Mean and Variance is interrelated. The first step is finding the mean which is done as follows, Mean = ( 610+450+160+420+310)/ 5 = 390 So the mean average is 390 mm. To calculate the Variance, compute the difference of each from the mean, square it and find then find the average once again. So for this particular case the variance is : = (2202 + 602 + (-230)2 +302 + (-80)2)/5 = (48400 + 3600 + 52900 + 900 + 6400)/5 Final answer : Variance = 22440 Problem & SolutionExample: Find the variance of the numbers 3, 8, 6, 10, 12, 9, 11, 10, 12, 7. Solution: Given, 3, 8, 6, 10, 12, 9, 11, 10, 12, 7 Step 1: Compute the mean of the 10 values given. Mean = (3+8+6+10+12+9+11+10+12+7) / 10 = 88 / 10 = 8.8 Step 2: Make a table with three columns, one for the X values, the second for the deviations and the third for squared deviations. As the data is not given as sample data so we use the formula for population variance. Thus, the mean is denoted by μ.
Step 3: \(\begin{array}{l}\sigma ^{2} = \frac{\sum (X-\mu )^{2}}{N}\end{array} \) = 73.6 / 10 = 7.36 Points to Remember
Note: If the data values are identical in a set, then their variance will be zero (0). Stay tuned with BYJU’S to learn more about Covariance Formula and other maths concepts with the help of interactive videos. Frequently Asked Questions – FAQsIn statistics, variance is a measure of spread of values or observations from mean. Variance is denoted by symbols: σ2, s2, or Var(X). The
formula to find the variance is given by: To find the variance easily, we need to find the
mean of given observations first. Then subtract this mean value from each of the observations and square them. At last, find the mean of the squared terms to get the variance. Variance is the square of standard
deviation for a given set of observations. If σ is the standard deviation then the variance is equal to σ2. What is a pattern of variation that is noted in a given data set?method. The pattern of variation in data is called the. distribution.
What best describes the basic steps of the scientific method?The scientific method has five basic steps, plus one feedback step:. Make an observation.. Ask a question.. Form a hypothesis, or testable explanation.. Make a prediction based on the hypothesis.. Test the prediction.. Iterate: use the results to make new hypotheses or predictions.. Which section of the research article includes an explanation of the procedures used to conduct the experiment?Abstract. The methods section of a research paper provides the information by which a study's validity is judged. Therefore, it requires a clear and precise description of how an experiment was done, and the rationale for why specific experimental procedures were chosen.
Which type of research observes and describes a large or small segment or segments of the environment at a single point in time?The descriptive research method primarily focuses on describing the nature of a demographic segment, without focusing on “why” a particular phenomenon occurs. In other words, it “describes” the subject of the research, without covering “why” it happens.
|