In statistics, a measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. As such, measures of central tendency are sometimes called measures of central location. They are also classed as summary statistics. The mean, median, and mode are the three most common measures of central tendency. Each of these measures provides a different indication of the typical or central value in the distribution of a set of numbers.
Understanding the central tendency of a dataset is crucial in drawing meaningful conclusions from the data. These measures are widely used in various fields, from business and economics to social sciences and medicine, to summarize and interpret data effectively.
The mean, or arithmetic average, is the most common measure of central tendency. It is calculated by summing all the values in a dataset and dividing by the total number of values.
For a dataset with n observations (x₁, x₂, …, xₙ), the formula for the sample mean (denoted as x̄) is:
x̄ = (Σx) / nWhere: * x̄ (pronounced “x-bar”) represents the sample mean. * Σx (the Greek letter sigma) indicates the sum of all values in the dataset. * n is the number of observations in the dataset.
For a whole population, the mean is denoted by the Greek letter μ (mu).
Let’s consider the following dataset representing the number of goals scored by a soccer team in 9 matches:
2, 4, 5, 5, 6, 7, 8, 8, 9
To find the mean, we first sum all the scores:
2 + 4 + 5 + 5 + 6 + 7 + 8 + 8 + 9 = 54
Next, we divide the sum by the number of matches (which is 9):
Mean = 54 / 9 = 6
So, the mean number of goals scored by the team is 6.
The mean is best used when the data is symmetrically distributed (not skewed) and does not have significant outliers (extreme values). In such cases, it provides a good representation of the central value.
The median is the middle value in a dataset that has been arranged in ascending or descending order. It is the point that divides the dataset into two equal halves.
(n + 1) / 2.n / 2 and (n / 2) + 1.Odd Number of Observations:
Consider the soccer team’s goals again:
2, 4, 5, 5, 6, 7, 8, 8, 9
The data is already ordered. With 9 observations (an odd number), the
median is the (9 + 1) / 2 = 5th value, which is
6.
Even Number of Observations:
Let’s add one more match with 10 goals:
2, 4, 5, 5, 6, 7, 8, 8, 9, 10
Now we have 10 observations (an even number). The two middle values
are the 10 / 2 = 5th value (6) and the
(10 / 2) + 1 = 6th value (7).
The median is the average of these two values:
(6 + 7) / 2 = 6.5