Measures of central tendency

A dataset’s central tendency can be summarised using a single value that represents the data’s distribution centre. A branch of descriptive statistics that includes both dataset variability (dispersion) and the central tendency is known as descriptive statistics. Measures of central tendency are referred to as such since they tell us what’s happening in the middle of the data.

To be more explicit, in many frequency distributions, the tabulated numbers show smaller frequencies at the beginning and end and bigger or higher frequencies at the middle of the distribution.

As a result, the variable’s typical values are clustered or grouped around the middle of the distribution. The central tendency (or location) of data refers to the tendency of data to be concentrated in the centre of the distribution.

The measure of central tendency listed below is:

There are a number of ways to measure a dataset’s central tendency, including:

• Mean: The sum of all the values in a dataset divided by the total number of values represents the mean (average).

• Median: The dataset’s median value is the midpoint in the ascending list of values (from the smallest value to the largest value). The median of a dataset is equal to the mean of the two middle values when the number of values in the dataset is even.

• Mode: The most often occurring value in a dataset is defined as the mode. It is possible to find many different modes in a dataset, but it is also possible to find none. (More of these are discussed below)

However, there are additional measures of central tendencies, such as the harmonic mean,  geometric mean, midrange, and a geometric median, which are less typically used. 

The characteristics of a dataset influence the measure of central tendency. For example, the mode is the only metric of central tendency for categorical data, although the median is best for ordinal data. 

Even though the mean is widely considered to be the best indicator of central tendency for quantitative data, this is not always the case. There are some datasets where the mean may not be a good fit because they contain huge or small numbers. The mean may be distorted by extreme values. As a result, additional options are on the table.

The best measure of central tendency can be derived mathematically or conceptually. A frequency distribution graph can also be used to identify them.

The types of arithmetic means

The two most common types of arithmetic mean are:

• Weighted arithmetic mean and

• Simple arithmetic mean

Simple Arithmetic means

• The arithmetic mean is the most widely used statistic for assessing the degree of central tendency.

• Each observation’s total numerical value is multiplied by the total number of observations to arrive at the total number of observations. 

• To put it another way, it is determined by summing up each observation’s value and then dividing the total by the number of observations in the set of observations.

Weighted Arithmetic means

• In the arithmetic mean, all of the objects or observations are given the same weight. All of these observations may not be of equal value or weightage, even if they are made in a sincere manner. It’s possible that some observations will be given more weight than others. In our family budget, for example, our everyday essentials (such as food) are given bigger weights than our luxuries.

• Weights are a way of describing the relative importance of various items by assigning numerical values to those values. Weighted AM is the mean stated as a function of the respective weights. Various economic problems are regularly studied using this type of metric.

Median

• An observation set’s median value is the midpoint of the range of values seen for that variable. For an odd number of observations, the (N+1/2)th value is used to find the middlemost value or the median of an ungrouped data set. 

• As a result, if there are an equal number of observations (N/2 and N+1/2), the median will be the average of the two middle values.

• Therefore, it’s clear that the median divides the data set in half. If an exceptionally big or tiny value is present, it does not affect the positional average. A grouped frequency distribution with open-ended classes can also be used to calculate it.

Median has its advantages

• The arithmetic mean loses some of its advantages when using the median instead. Using the arithmetic mean to get the average of a big or small group of observations might be misleading. We already know that the per capita income of an Indian is lower than the per capita income of a US citizen. The per capita incomes of wealthy and impoverished Indians are similar.

• In terms of income, the rich and the poor are vastly different. India’s per capita income will rise if the wealthier sections of society see an increase in income, but the poor do not. Even yet, it’s difficult to label someone with such a high median income as a true reflection of the population. In this case, the median value of income is a stronger indicator of per capita income than the sum of the individual values.

Mode

Another useful statistical technique for determining a variable’s central tendency is the mode. In the case of a collection of observations on a discrete variable, the mode refers to the observation with the highest frequency. The most common value in a set of numbers is the mode, which is the value that appears the most frequently.

How are mean, median, and mode inter-related?

The frequency distribution of a series of observations reported on a variable can be symmetrical or normal, or asymmetrical, with the mean, median, and mode all in the same place. This is known as a symmetrical distribution.

As a result of this, a relationship known as the Karl-connection Pearson’s is noticed and established in skewed distributions as follows: 

• Mean – Mode = 3 (Mean – Median)

Conclusion

We have studied here the different measures of central tendency and its different characteristics. We have also determined the relationships between them. 

It is safe to conclude that selecting a single measure of central tendency from the three provided is not a good idea in all instances. Actually speaking, it is a crucial responsibility for the user or investigator and researcher to select the most suitable or ideal one to fit his or her own purpose, depending on the kind and quality of the data provided and the study’s objective.