Before we can determine the variance of the set, we must first determine the value of x, our unknown. The problem tells us the mean of the set, and we know each number in the set except for x, so we can solve for x as follows:

\(\displaystyle \mu =\frac{\sum_{i=1}^{N}x_i}{N}=\frac{37+42+39+33+x}{5}=40\)

\(\displaystyle 151+x=200\rightarrow x=49\)

Now that we know all the values of the set, the mean, and the number of values, we can solve for the variance using the following equation:

\(\displaystyle \sigma ^2=\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu )^2\)

Where \(\displaystyle \sigma ^2\) is the variance, N is the number of values in the set, \(\displaystyle x_i\) is the value currently being evaluated in the set, and \(\displaystyle \mu\) is the mean of the set. We start by determing the value of each term in the summation part of the formula:

\(\displaystyle (37-40)^2=9\)

\(\displaystyle (42-40)^2=4\)

\(\displaystyle (39-40)^2=1\)

\(\displaystyle (33-40)^2=49\)

\(\displaystyle (49-40)^2=81\)

Now that we now each value of the summation, all we have left to do is add them all together and divide the final result by N:

\(\displaystyle \sigma ^2=\frac{1}{5}(9+4+1+49+81)=\frac{144}{5}=28.8\)

Write the formula for variance.

\(\displaystyle \sigma^2 = \frac{\sum (X-\mu)^2 }{N}\)

The value of \(\displaystyle X\) represents each number in the data set.

The value of \(\displaystyle N\) is 3, since there are 3 numbers in the data set.

Find the mean, \(\displaystyle \mu\).

\(\displaystyle \mu= \frac{2+3+4}{3} = \frac{9}{3}=3\)

Rewrite the variance formula:

\(\displaystyle \sigma^2 = \frac{(2-3)^2 +(3-3)^2+(4-3)^2}{3} = \frac{1+0+1}{3}=\frac{2}{3}\)

Write the formula relationship between standard deviation and variance.

The standard deviation is the square root of variance.

\(\displaystyle \sigma = \sqrt{\textup{Variance}}\)

Substitute the standard deviation.

\(\displaystyle 8.6 = \sqrt{\textup{Variance}}\)

Square both sides.

\(\displaystyle \textup{Variance} =( 8.6)^2 = 73.96\)

The answer is:  \(\displaystyle 73.96\)

Write the formula for the variance.

\(\displaystyle \textup{Variance} =\sigma^2\)

\(\displaystyle \textup{Standard Deviation}= \sigma\)

The standard deviation is the square root of variance.

\(\displaystyle \sigma = \sqrt{\textup{Variance}}\)

Substitute the value of standard deviation in the formula.

\(\displaystyle 19.5 = \sqrt{\textup{Variance}}\)

Square both sides.

\(\displaystyle \textup{Variance }= 380.25\)

The answer is:  \(\displaystyle 380.25\)

The variance of a distribution is equal to the standard deviation squared.

\(\displaystyle 9^2=81\)

To determine the variance, we will need to find the squared differences of the mean, and take the average.

Find the mean.

\(\displaystyle \frac{5+7+27}{3} = \frac{39}{3}=13\)

Subtract each number from the mean and square the quantities.

\(\displaystyle (5-13)^2+(7-13)^2+(27-13)^2 = 64+36+196 = 296\)

Divide this value by three.

The answer is:  \(\displaystyle \frac{296}{3}\)

To get the standard deviation, we need to calculate the variance, which is the average of the squared differences from the mean, so we will start by getting the mean.

\(\displaystyle \frac{(2+3+3+4+5+7)}{6}=4\)

Then, subtract the mean from each value...

\(\displaystyle (2-4)^{2}=(-2)^{2}=4\)

\(\displaystyle (3-4)^{2}=(-1)^{2}=1\)

\(\displaystyle (4-4)^{2}=(0)^{2}=0\)

\(\displaystyle (5-4)^{2}=(1)^{2}=1\)

\(\displaystyle (7-4)^{2}=(3)^{2}=9\)

...and take the mean of these resulting values, which is equal to the variance.

\(\displaystyle \frac{(4+1+1+0+1+9)}{6}=2\frac{2}{3}=\frac{8}{3}\)

The square root of this value is the standard deviation. The answer is presented as \(\displaystyle \sqrt{\frac{8}{3}}\),

but you may also calculate it and find it equal to about \(\displaystyle 1.63\)

Formula for the standard deviation:

\(\displaystyle \sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}\)

1. Find the mean

\(\displaystyle \frac{12+15+30+5+27+19}{6}=\frac{108}{6}=18\)

2. Subtract the mean from each number in the data set

\(\displaystyle (12-18)^{2}=(-6)^{2}=36\)

\(\displaystyle (15-18)^{2}=(-3)^{2}=9\)

\(\displaystyle (30-18)^{2}=(12)^{2}=144\)

\(\displaystyle (5-18)^{2}=(-13)^{2}=169\)

\(\displaystyle (27-18)^{2}=(9)^{2}=81\)

\(\displaystyle (19-18)^{2}=(1)^{2}=1\)

3. Sum up the square of the differences and divide by n

\(\displaystyle \frac{36+9+144+169+81+1}{6}=\frac{440}{6}=\frac{220}{3}\)

4. Take the square root of the variance

\(\displaystyle \sqrt{\frac{220}{3}}= 8.56\)

The 68-95-99.7 rule states that nearly all values lie within 3 standard deviations of the mean in a normal distribution. In this case the question asks for 95% so we want to know what 2 standard deviations from the mean is.

We are given the variance, so to find the standard deviation, take the square root.

\(\displaystyle \sqrt{16}=4\)

So two standard devations is 8 inches. 95% of heights should be within 8 inches of the mean.

The standard deviation of a set of numbers is how much the numbers deviate from the mean. More formally, the standard deviation is 

\(\displaystyle \sigma =\sqrt{\frac{\sum{(x-M)^2} }{N-1}}\)

where \(\displaystyle x\) is a number in the series, \(\displaystyle M\) is the mean, and \(\displaystyle N\) is the number of data points. So, to calculate the standard deviation, we must first calculate the mean. The mean of this data set is

\(\displaystyle M = \frac{85+75+97+83+62+75+88+84+92+89}{10}=83\)

Now that we know the mean, we can start calculating the standard deviation. We first need to find the sum of each data point minus the average squared.

\(\displaystyle (85-83)^2+(75-83)^2+(97-83)^2+(62-83)^2+(75-83)^2+(88-83)^2+(84-83)^2+(92-83)^2+(89-83)^2\)

Calculating that, we get that the variance from the mean is \(\displaystyle 912\). Plugging that into our equation for standard deviation, with \(\displaystyle N\) being ten data points, we get

\(\displaystyle \sigma = \sqrt{\frac{912}{9}}=10.07\)