Variance and Standard Deviation: Formula & Properties

Ans. All possible outcomes of rolling a die are {1; 2; 3; 4; 5; 6}.

Therefore, (n) = 6

⇒ Before finding the variance, we need to find the mean of the data set.

⇒ Mean = (1+2+3+4+5+6)/6 = 3.5

⇒ We can put the value of data and mean in the formula to get;

⇒ σ² = [(1-3.5)² + (2-3.5)² + (3-3.5)² + (4-3.5)² + (5-3.5)² + (6-3.5)2 ]/6

⇒ σ² = \(\frac{(6.25+2.25+0.25+0.25+2.25+6.25)}{6}\)

⇒ Variance (σ² ) = 2.917

Now, Standard Deviation (σ) = √ (σ² )

⇒ Standard Deviation (σ) = √(2.917) 

∴ Standard Deviation (σ) = 1.708

Ans. Key differences between variance and standard deviation are as listed below.

• Calculation: Standard deviation is the square root of variance. Variance is computed as the average of the squared differences between each data point and the mean.
• Interpretation: Standard deviation provides a measure of the typical distance between data points and the mean, while variance quantifies the average squared distance between data points and the mean.
• Units: Standard deviation shares the same units as the original data, whereas variance is in units squared.
• Representation: Standard deviation is symbolised by σ for populations and s for samples, while variance is represented by σ² for populations and s² for samples.

Ans.To calculate the variance and standard deviation for the given data {2, 5, 8, 2, 5}, follow these steps:

⇒Calculate the mean: Mean = 2+5+8+2+55 = \(\frac{22}{5}\)= 4.4

⇒ Calculate the squared differences from the mean: Subtract the mean from each data point and square the result.

⇒ (2−4.4)²=5.76

⇒ (5−4.4)²=0.36

⇒ (8−4.4)²=12.96

⇒ (2−4.4)²=5.76

⇒ (5−4.4)²=0.36

⇒ Variance=5.76+0.36+12.96+5.76+0.36/5

∴ Variance = 5.04

∴ Standard Deviation=5.04 ≈ 2.24

Ans. To compute the variance,

• Calculate the Mean: Add up all the values in the dataset and divide by the total number of values. It gives the mean or average of the dataset.
• Calculate the Squared Differences from the Mean: Subtract the mean from each data point in the dataset, then square the result. It gives the squared differences from the mean for each data point.
• Find the Average of the Squared Differences: Add up all the squared differences calculated in the previous step, then divide by the total number of data points. It gives the variance of the dataset.

Ans. Variance is important because it measures the spread or dispersion of data points in a dataset, providing valuable insights into the variability of the data. It helps in understanding the distribution, assessing risk, comparing datasets, and making informed decisions in various fields such as finance, quality control, and statistical analysis.

1. S²
2. S
3. $
4. b²

Ans. a. S²

Explanation: symbol of variance in a population is σ² and sample is S². Therefore, option a is the correct answer.

Ans. Since mean and variance is interrelated. 

⇒ First calculate the mean which is as follows:

⇒ Mean = \(\frac{( 10 + 20 + 50 + 40)}{ 4}\) = 30

So the average mean is 30 m.

⇒ To calculate the variance, compute the difference of each from the mean.

Then square it.

⇒ Lastly, find the average once again.

So for this particular case the variance is :

= ((-25)² + (-10)² + (20)² +(10 )² / 4

= \(\frac{(625 + 100 + 400 + 100)}{4}\)

∴ Variance = 306.25

Ans. When you roll a die then there are six possible outcomes which go as {1,2,3,4,5,6} and total number of samples is 6. 

⇒To find the variance first calculate the mean of the data set.

⇒ Mean, x̅ = \(\frac{(1+2+3+4+5+6)}{6}\) = 3.5

⇒ Now put the value of data and mean in the variance formula

⇒ σ² = Σ (xi – x̅)²/n

⇒ σ² = ⅙ (6.25+2.25+0.25+0.25+2.25+6.25)

∴ σ² = 2.917

Ans. Since mean and variance is interrelated. 

⇒ First calculate the mean which is as follows:

⇒ Mean = \(\frac{( 90 + 40 + 50 + 60)}{ 4} \)= 60

So the average mean is 60 m.

⇒ To calculate the variance, compute the difference of each from the mean.

Then square it.

⇒ Lastly, find the average once again.

So for this particular case the variance is :

= ((30)² + (-20)² + (-10)² +(0 )² )/ 4

= \(\frac{(900 + 100 + 400)}{4}\)

∴ Variance = 350 and standard deviation = 18.7

Ans. Even Numbers less than 8 are {0, 2, 4, 6}

⇒ This data set has four values (n) = 4

⇒ First calculate the mean of the data set.

⇒ Mean, x̅ = \(\frac{(0+2+4+6)}{4}\) = 3

⇒ Put the value of data and mean in the variance formula: σ² = Σ (xi – x̅)2/n

⇒ σ² = [(0-3)² + (2-3)² + (4-3)² + (6-3)²]/3

⇒ σ² = (9 + 1 + 1 + 9)/3 = 20/3

∴ Variance (σ²) = 6.67

Now, Standard Deviation (σ) = √ (σ²)

⇒ Standard Deviation (σ) = √(6.67) 

∴ Standard Deviation (σ) = 2.58

Ans. Since mean and variance is interrelated. 

⇒ First calculate the mean which is as follows:

⇒ Mean = \(\frac{( 100 + 40 + 50 + 60 + 30)}{ 5}\) = 56

So the average mean is 56 m.

⇒ To calculate the variance, compute the difference of each from the mean.

Then square it.

⇒ Lastly, find the average once again.

So for this particular case the variance is :

= ((44)² + (-16)² + (-6)² +(4 )² + (26)² )/ 4

= \(\frac{(1936 + 256 + 36 + 16 + 676)}{4}\)

∴ Variance = 730 and standard deviation = 27.01