Question

Find the variance and standard deviation of the set of data to the nearest tenth (one decimal place){4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 9}

Answer 1

Hello there. To solve this question, we'll need to remember how to find the variance and standard deviation of a data set.

First, given a data set with values:

`\mleft\lbrace x_1,x_2,x_(3,\ldots)\mright\rbrace`

The variance can be calculated by the formula:

`\sum ^{}_{}((x_i-\mu)^2)/(n)`

In this case, μ is the arithmetic mean of the data set and x_i is the i-th element of the data set.

The data set given is:

{4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 9}

To calculate the mean, we add every term and divide by the number of terms:

`(4+5+5+5+5+5+6+6+6+6+7+7+7+7+8+9)/(16)=(98)/(16)=6.125`

Now, we calculate the square of the difference between every term and mean.

Notice we have some repeated terms, we rewrite the sum like this:

`((4-6.125)^2+5\cdot(5-6.125)^2+4\cdot(6-6.125)^2+4\cdot(7-6.125)^2+(8-6.125)^2+(9-6.125)^2)/(16)`

Adding the terms and calculating the squares, we have:

`\begin{gathered} ((-2.125)^2+5\cdot(-1.125)^2+4\cdot(-0.125)^2+4\cdot(0.875)^2+(1.875)^2+(2.875)^2)/(16) \\ \\ (4.515625+6.328125+0.0625+3.0625+3.515625+8.265625)/(16) \\ \\ (25.75)/(16)\text{ = }1.609 \end{gathered}`

This is the variance of the values of this data set. We can round it up to the nearest tenth as 1.6

The standard deviation is the square root of the variance.

Then, we calculate:

`\sigma\text{ = }\sqrt[]{1.6}=1.268`

Again, rounding it to the nearest tenth, we have 1.3;

The final answer is: The variance of this data set is 1.6 and its standard deviation is 1.3.

Solving the second question:

We apply the same formula. First, find the mean of the values:

`\mu\text{ = }(4.3+6.4+2.9+3.1+8.7+2.8+3.6+1.9+7.2)/(9)=4.54`

Now, as every term is different from each other, we apply the formula and get the following:

`((4.3-4.54)^2+(6.4-4.54)^2+(2.9-4.54)^2+(3.1-4.54)^2+(8.7-4.54)^2+(2.8-4.54)^2+(3.6-4.54)^2+(1.9-4.54)^2+(7.2-4.54)^2)/(9)`

Calculating the difference, squaring it, adding the values and dividing by 9, we get:

Variance is approximately equal to 4.84

The Standard deviation is the square root of the variance, namely 2.2