Solution - Statistics

Divide the sum by the number of terms:

Sum
$$21,760$$
Number of terms
$$4$$

$$x̄=5,440=5,440$$

The mean equals $$5,440$$

Arrange the numbers in ascending order:
256,1024,4096,16384

Count the number of terms:
There are (4) terms

Because there is an even number of terms, identify the middle two terms:
256,1024,4096,16384

Find the value that is halfway between the middle two terms by adding them together and dividing by 2:
$$\left(1024+4096\right)/2=5120/2=2560$$

The median equals 2,560

To find the range, subtract the lowest value from the highest value.

The highest value equals 16,384
The lowest value equals 256

$$16384-256=16128$$

The range equals 16,128

To find the sample variance, find the difference between each term and the mean, square the results, add together all of the squared results, and divide the sum by the number of terms minus 1.

The mean equals 5,440

To get the squared differences, subtract the mean from each term and square the result:

To get the sample variance, add together the squared differences and divide their sum by the number of terms minus 1

Sum:
$$119771136+1806336+19501056+26873856=167952384$$
Number of terms:
$$4$$
Number of terms minus 1:
3

Variance:
$$\frac{167952384}{3}=55984128$$

The sample variance ($$s^2$$) equals 55,984,128

The standard deviation of the sample equals the square root of the sample variance. This is why the variance is usually represented by a squared variable.

Variance: $$s^2=55,984,128$$

Find the square root:
$$s=\sqrt{\left(55984128\right)}=7482.254$$

The standard deviation ($$s$$) equals 7482.254