The variance of 20 observations is 5. If each observation is multiplied by 2, find the new variance of the resulting observations.

Step 1: Understand the given information

We are given that there are 20 observations, and their variance is 5. We need to find the new variance if each observation is multiplied by 2.

Step 2: Recall the property of variance

A key property of variance states that if each observation in a dataset is multiplied by a constant $k$, the new variance will be $k^2$ times the original variance. This is because variance measures the spread of data, and multiplying by $k$ scales the spread by $k$, so the squared spread (variance) scales by $k^2$.

Step 3: Identify the constant multiplier

In this problem, each observation is multiplied by 2. Therefore, the constant multiplier $k$ is 2.

Step 4: Calculate the new variance

Using the property of variance, we substitute the value of $k=2$ and the original variance of 5 into the formula. This gives us $2^2 \times 5$, which simplifies to $4 \times 5$, resulting in a new variance of 20.

More from this chapter

• Calculate the mean and variance of the first n natural numbers.
• For a statistical data x_1, x_2,, x_10 of 10 values, a student obtained the mean as 5.5 and Σ_i=1^(10) x_i² = 371. He later found that he had noted two values i
• The mean and standard deviation of 100 observations were calculated as 40 and 5.1, respectively by a student who took by mistake 50 instead of 40 for one observ
• Let the mean and variance of five observations x_1=1, x_2=3, x_3=a, x_4=7 and x_5=b, with a>b, be 5 and 10 respectively. Then the variance of the observations n
• 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.
• If each of the observation x_1, x_2,..., x_n is increased by ' a ', where a is a negative or positive number, show that the variance remains unchanged.
• The variance of the numbers 8, 21, 34, 47,, 320 is:
• Find the mean deviation about the median for the following data: x_i: 3, 6, 9, 12, 13, 15, 21, 22 f_i: 3, 4, 5, 2, 4, 5, 4, 3.

All Statistics solutions →