In a Nutshell: Central Tendency

• The mean, median, and mode are the three measurements of central tendency.

• The most often used measure of central tendency is the arithmetic mean.

• If the data is skewed, the median should be used as the measure of central tendency.

Central tendency is defined as

Central tendency measures are a summary measure that characterises a data set with a single number that reflects the middle of the distribution. The three most prevalent metrics of central tendency are as follows:

• The mean represents the data set's average.

• The median is the value in the middle.

• Mode- The most often occurring value in a data set.

It is also critical to understand measures of variability when performing descriptive statistics. You can also describe the distribution of the data set to summarise it.

Central tendency: Distributions

A data set is defined in statistics as a distribution of n values or scores.

The standard distribution

The data in a normal distribution is distributed symmetrically. In this situation, the mean, median, and mode values would all be the same. An example of a regularly distributed data set is as follows:

Distributions that are skewed

More values will fall on one side of the centre than the other in a skewed distribution. In such instances, the mean will be higher than the median, which will be higher than the mode.

The mode would be bigger than the median in a negatively skewed distribution, and the mean would be less than both of these numbers.

Moderate central tendency

The mode is the most commonly occurring value in a distribution. To locate the mode, arrange the data in ascending or descending order, and then look for the middle value.

You may obtain one mode, numerous modes, or no mode at all depending on the nature of the data set. The mode is the variable having the highest frequency in a frequency table.

If you wish to utilise a bar graph, simply look at the highest bar, which displays the mode. Consider the following example:

In this situation, the mode is 6 because this is the most common shoe size recorded.

When to use the mode

Because nominal data is categorised into mutually exclusive categories, mode is typically utilised with it.

Because you will be dealing with several variables while dealing with ratio data, it is not required to use the mode. Consider the following ratio data:

Median central tendency

The median is the middle value in a data set, and it can be found by sorting the data in ascending or descending order.

You can see that the precise middle point is at $4,001-$6,000 by arranging the data from low to high.
 

Median of an odd-numbered data set

In an odd-numbered data set, locate the value   at the Central tendency median location to get the median. The 'n' in the formula stands for the number of values in the data set. Because the total number of values in the preceding example is 33, you can use the following formula:

You may find the median by finding the value in the 17th place.

Median of an even-numbered data set

If the data set contains an even number of variables, you must discover the   and values for the Central tendency median even data set and the Central tendency median even data set 1. After that, add the two numbers together and divide them by two. The median of a data collection with 60 values is the mean of the values at these positions:

and 

Mean central tendency

The most often used measure of central tendency is the arithmetic mean. It represents the data set's average and is calculated by summing all of the values and dividing the result by the number of values. The geometrical mean, on the other hand, is determined as the n-root of the product of all the values. The arithmetic mean of the data set (3,4,6,8,14) can be found by summing all of the values. Divide this number by n, which in this case is 5, to determine the mean. 

The impact of outliers on the mean

Data outliers are values that deviate significantly from the rest of the values in a data set. These values can dramatically raise or lower the mean compared to the other values. For example, in the data set (3,5,7,9,300), the mean is 64.8, which does not adequately represent the data set. 
 

Mean population vs. sample

The mean of a sample or a population can be calculated. The population and sample means are calculated in the same manner, but the notations change. The 'n' sign, for example, denotes the number of variables in the sample data set, while the 'N' symbol represents the population. 

Central tendency – Mean, median, or mode?

Because they have various strengths and limits, all three measures of central tendency should be employed together. However, in other circumstances, using one or two measures of central tendency may be insufficient.

• Although the mode can be applied to all four levels of measurement, it is most commonly utilised with nominal and ordinal data.

• The median can only be applied to ordinal, ratio, and interval data.

• The mean can only be used with interval or ratio measurement levels.

When selecting a measure to utilise in a certain data collection, you must examine the data distribution. If the distribution is regularly distributed, you can use mean, median, or mode because they all have the same value. The median should be used for skewed data.

Frequently asked questions

What are the central tendency measures?

The mean, mode, and median are examples of central tendency measures.

What is the best measure of central tendency to use in a strongly skewed distribution?

If the distribution is strongly skewed, you should use the median.

Can I use measures of central tendency on all levels of data?

You can use mode on all levels of data, but median and mean cannot be used on nominal data.

When is mode the best measure of central tendency?

Mode is preferred when dealing with nominal data.