Statistical Measures

Mean

The mean, also called the mathematical average of a given data set, can be found by adding all values in the data set, and dividing by the number of values. We can use a mathematical formula to describe this: \(\mu = \frac{\Sigma x}{n}\) where \(\mu\) is used to represent the mean.

Mode

The mode is the most frequently occurring value in a data set. Sometimes you will have a data set where this describes more than one value. Here they are all considered the mode.

Median

The median is the midpoint value of a given data set. In cases where the midpoint values are two (when the number of data points is even), you need to find the average of both middle values. When finding the median, it is appropriate to reorder your values in ascending order. Take the \(\frac{n+1}{2}\) value if the number of data points is odd. When the number is even, take the \(\frac{n}{2}\) and the value \(\frac{n+2}{2}\).

What are measures of spread?

Measures of spread are statistical measures that describe the similarity and variety of the values of given datasets. Relying on central tendency measures alone as a summary description for data sets can be very misleading since it does not account for extreme values. Measures of spread help us do that, including range, variance, and standard deviation.

Range

The range is the difference between a given data set's highest and lowest values. It helps you to know how wide the data is. To find the range, the lowest value in the data is subtracted from the highest value.

However, the range has a few limitations:

• It is affected by outliers.

• It cannot be used for open-ended distribution.

Quartiles and the interquartile range

A quartile is a type of quantile that divides an ordered data set into four parts (quarters). A quartile is not the group of numbers that have been divided. It is the cut-off point in the division.

The interquartile range is the difference between the upper quartile and the lower quartile value.

To find the quartile of a given data set you can proceed as follows:

1. Order the values in ascending order.

2. Find the median. This is always labelled as the second quartile (Q₂).

3. Now find the median of both halves of the data set. The lowest half is labelled Q₁, and the highest half is labelled Q₃.

4. Find the interquartile range (IQR) by subtracting Q₁ from Q₃.

Variance and standard deviation

Variance and standard deviation are both measures of variability. The variance is the measure of how data points vary from the mean, and the standard deviation is the square root of variance. What this tells us is that standard deviation is derived from variance.

Variance is denoted by \(\sigma^2\)

Standard deviation is denoted by \(\sigma\).

Variance formula

The population variance formula is \(\sigma^2 = \frac{\Sigma(x_1 - \mu )^2}{N}\)

Where \(\sigma^2\) = population variance

N = size of the population

x_i = each value from the population

\(\mu\) = the population mean.

The sample variance formula is \(s^2 = \frac {\Sigma(x_i - \bar{x})^2}{n-1}\)

Where s² = sample variance

n = size of sample

x_i = each value from the sample

\(\bar{x}\) = the sample mean.

Standard deviation formula

The population standard deviation formula is given by \(\sigma = \sqrt{\frac{\Sigma(x_i - \mu)^2}{N}}\)

Where \(\sigma\) = population standard deviation.

N = size of the population.

x_i = each value from the population.

\(\mu\) = the population mean.

The sample standard deviation formula is given by \(s = \sqrt {\frac{\Sigma(x_i - \bar {x})^2}{n-1}}\)

Where s = sample standard deviation.

n = size of sample.

x_i = each value from the sample.

\(\bar{x}\) = the sample mean.