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Difference between Variance and Standard Deviation is that they are two different types of absolute measures of variability that describe how the data is distributed around the mean. The average of the squares of the deviations is the variance. Standard deviation, on the other hand, is the square root of the numerical value obtained when computing the variance.
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Key Terms: Variance, Standard Deviation, Average Deviation, Arithmetic Mean, Data Sets, Mean, Median, Mode
What is Variance?
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Variance is defined as the difference between two or more values. The measure of variability that represents how far members of a group are spread out is known as variance in statistics. It determines the average deviation of each observation from the mean.
Distribution of Variance
When the variance of a data set is low, it indicates that the data points are close to the mean, whereas a high variance indicates that the observations are widely dispersed around the arithmetic mean and from one another.
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Formula of Variance
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Variance is given by the formula,
| σ2 = ∑ (x – M)2/n |
Where,
- M → Mean
- X → Data set's value.
- N → Number of observations in a data set.
What is Standard Deviation?
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Standard deviation is a metric that expresses how dispersed the observations in a dataset are. A low standard deviation indicates that the scores are close to the arithmetic mean, whereas a high standard deviation indicates that the scores are dispersed across a wider range of values.
Standard Deviation
Also Check: Population Mean Formula
Formula of Standard Deviation
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Standard Deviation is given by the formula,
| σ = √∑ (x – M)2/n |
Where,
M → Mean
X → Data set's value.
N → Number of observations in a Data Set
Coefficient of Variation Detailed Explanation
Key Differences Between Variance and Standard Deviation
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The key differences between Variance and Standard Deviation are tabulated below.
| Variance | Standard Deviation |
|---|---|
| Average squared deviations from the mean | The variance's square root. |
| It is denoted by σ2. | It is denoted by σ. |
| Measures the data set's dispersion. | It is used to calculate the spread around the mean. |
| Variance does not have a sub-additive relationship. | For symmetrical distributions with no outliers, this is a measure of spread. |
| Variance is also a measure of a population's data volatility. | In finance, the standard deviation is commonly referred to as volatility. |
| The variance of an outcome is a measure of how far it deviates from the Mean. | The standard deviation is a measurement of how far the expected value differs from the normal standard deviation. The standard deviation can be used to quantify uncertainty. |
| It aids in determining the actual deviation of performance from the standard in finance. | Because it measures the risk associated with Market Volatility, Standard Deviation is a useful tool for making investment decisions in stocks, mutual funds, and other securities. |
| Knowing the Variance allows you to take corrective action. | The process of identifying, measuring, and mitigating the risks associated with a project, investment, or business is known as risk analysis. Quantitative and qualitative risk analysis are the two types of risk analysis.” It is the analysis and interpretation of the data gathered during the calculation of the standard deviation of various stocks, and the data is analyzed in order to make an informed investment decision. |
Applications of Variance and Standard Deviation
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- To quantify the measure of spread, the variance combines all of the values during a set of knowledge.
- The larger the range, the more the variation ends when the values within the data set are separated by a larger gap.
- The statistical probability distribution of living volatility from the mean is known as variance, and volatility is one of the risk analysis measures that can help investors assess the risk in their investment portfolios. It is also an important aspect of asset allocation.
Standard Deviation and Variance used in Finance
- The variance, on the other hand, is widely used in a variety of fields, including finance, as a measure of market and security volatility. These values are also required for dog routes, class strength, weather forecasting, class-by-class student appearance in exams, department-by-department salary checks, and market research.
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Things to Remember
- The measure of variability that represents how far members of a group are spread out is known as variance in statistics.
- Variance is given by the formula σ2 = ∑ (x – M)2/n
- The standard deviation is a metric that expresses how dispersed the observations in a dataset are.
- Standard Deviation is given by the formula σ = √∑ (x – M)2/n
- The spread of data from its mean point is measured by both variance and standard deviation.
Previous Year Questions
- Find the Variance of the given table … (BITSAT 2013)
- The Variance of first 50 even natural Numbers … (JEE Mains 2014)
- Standard Deviation of first n odd numbers … (KEAM)
- The mean deviation from the median of the following set of observations 5, 3, 9, 12, 3, 10, 12, 21, 18, 12, 21 is
- The mean marks of 120 students is 20. It was later discovered that two marks were wrongly taken as 50 and 80 instead of 15 and 18. The correct mean of marks is
- The mean of 100 observations is 50 and their standard deviation is 55. The sum of all squares of all the observations is
- The mean of 12 numbers is 24. If 5 is added in every number, the new mean is
- The mean of 5 observations is 4.4 and their variance is 8.24. If three of the observations are 1, 2 and 6, find the other two observations.
- The mean of 50 observations is 36. If two observations 30 and 42 are deleted, then the mean of the remaining observations is
- The mean of six numbers is 30. If one number is excluded, the mean of the remaining numbers is 29. The excluded number is
- The mean of the first n terms of the A.P. (a + d) + (a + 3d) + (a + 5d) + .... is
- The mean of the given data is 30. If total data is 70, then missing numbers are
- The mean of two samples of size 200 and 300 were found to be 25, 10 respectively. The variance of combined sample of size 500 is
- The mean of combined group of 10 + n items is found to be 30. The value of n is
Sample Questions
Ques. What are the meanings of mean and standard deviation? (2 marks)
Ans. Data are clustered around the mean when the standard deviation is low, and data are more spread out when the standard deviation is high. A standard deviation near zero indicates that data points are close to the mean, while a high or low standard deviation indicates that data points are above or below the mean, respectively.
Ques. Find the standard deviation of the first n natural numbers. (5 marks)
Ans. 
Ques. In research, what is the significance of mean-variance and standard deviation? (2 marks)
Ans. Standard deviation and variance, at their most basic level, put scores into context. Knowing the mean and standard deviation on any given exam, for example, allows students to assess how well they performed in comparison to other students in the class.
Ques. Two sets each of 20 observations have the same standard derivation 5. The first set has a mean of 17 and the second a mean of 22. Determine the standard deviation of the set obtained by combining the given two sets. (3 marks)
Ans. 
Ques. What is a standard deviation and what does it mean? (2 marks)
Ans. The standard deviation indicates how dispersed the data is. It's a metric for determining how far each observed value deviates from the mean. Approximately 95% of values in any distribution will be within 2 standard deviations of the mean.
Ques. Find the standard deviation. (5 marks)
Ans. 
Ques. Why does the Standard Deviation Value always have a positive value? (2 marks)
Ans. So, the standard deviation will never be negative again. Because the standard deviation equation's denominator and numerator are both positive, it is derived from squared values. If you take any data and calculate the variance, then calculate the standard deviation, the resultant value is always in the + sign, according to the equations.
Ques. The sum of 10 values is 100 and the sum of their squares is 1090. Find out the coefficient of variation. (2 marks)
Ans. 
Ques. What do you mean when you say "Deviation"?(2 marks)
Ans. Deviation distinguishes things from one another in different situations. In statistics, the deviation is defined as the distance between values and the mean in a given data set. The distinction is useful in a variety of fields, including business, finance, and statistics.
6Ques. The mean life of a sample of 60 bulbs was 650 hours and the standard deviation was 8 hours. A second sample of 80 bulbs has a mean life of 660 hours and a standard deviation of 7 hours. Find the overall standard deviation. (3 marks)
Ans. 
Ques. What is the relationship between variance and standard deviation? (2 marks)
Ans. The square root of the variance is used to calculate standard deviation, which measures how far a group of numbers is from the mean. The variance is a measurement of how far each data point deviates from the mean (the average of all data points).
Ques. Mean and standard deviation of 100 items are 50 and 4, respectively. Then find the sum of all the items and the sum of the squares of the items. (3 marks)
Ans. 
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