Central Measures

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Central Value

It is often preferable to use a few numbers in order to summarize a distribution. One crucial context of distribution is where its centre is positioned. In this aspect, measures of central tendency are discussed primarily. A second important aspect of a distribution is how expanded it is. Simply to say, how much the data in the distribution differ from each other. Thus, a measure of central tendency represents the central value (centre point value) or typical value of a dataset.


Uses of Central Measures

Central tendencies are useful any time you want to summarize the central location of a dataset using a single value.


Finding a Central Value

The three most common measures of central tendency are as follows:

  1. The mean

  2. Median

  3. Mode

Each of these measures computes the location of the central point using a different technique. Furthermore, these measures help to find out where most values in a distribution fall and are also known as the central location of a distribution.


Mean

The mean is the arithmetic average, and it is possibly the measure of central tendency that most of us are familiar with. The computation of the mean or average executes all values in the data. If you would change any value, the mean will also change. However, the mean doesn’t always trace the centre of the data or the central value correctly. The mean can be displayed in distributions.


Median

Median can be used for different measures of variability. If you’re not taking into account the mean because your data is skewed, you will see that using the median for the central tendency and interquartile range (IQR) for the variability goes together wonderfully. The median divides that data in half and the interquartile range tells you where the middle half of the data fall. The wider the IQR, the greater the spread the data spread. You can also use percentiles in order to identify the spread for other proportions. For example, 95% of the data fall between the 2.5th and 97.5th percentiles.


Mode

The mode is actually the component of the data set that occurs most frequently. One way to handle data spread when not using the mean is the mode. Mode for continuous data would be important when it comes to distributions that have two (bimodal) or more (multi-modal) peaks. In these cases, where one has more than one centre of tendency, it would seem that the mode measure of central tendency becomes the more important piece of information than either the mean or median.


What is Central Value?

When we define a central value, we state it to be a single value that attempts to define a data set by determining the central position within that set of data. Seeing that, a measure of central tendency is sometimes referred to as measures of central location or summary statistics.


Fun Facts on Central Value Statistics

  • By looking only on the mean, you don’t know how far from the mean any given observation is possible to fall.

  • If you knew an individual was male or female, those subpopulations will likely have different means.

  • The measure of central tendency cannot give a complete picture of data for interpretation.

Solved Examples on Central Measures

Example:

Palak's exam scores for her last chemistry class were: 67, 53, 74, 78. Find out the mean of these values.

Solution:

List the values of data set in order and divide the sum total of all values by the number of values in the data. Doing so, we get:

53 + 67 + 74 + 78 / 4

= 272/4

= 68

Note: if we get an answer in decimal form, then we round to one more decimal place than the initial data had.


Example: Find the median of the fluctuation in prices of onion: 4, 6, 3, 10, 8, 2, 5, 7, 6, 4, 8.

Solution:

Start by arranging the data in ascending order: 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, and 10.

Seeing that there are 11 data values, an even number, thus our middle number is 6.

Now, for finding a central value of the two middle numbers is 6 (6th value of the data set).

Thus, the median fluctuation in onion price is 6.


Conclusion: Although Mean, Median and Mode are the most commonly used methods of finding the central value of a number, there are other ways too like harmonic mean, geometric mean.

FAQs (Frequently Asked Questions)

Q1. How do we Find the Most “Typical” Central Value of a Data Set?

Answer: Focus on the term that we have used here i.e “typical”. Though in many instances, you might think of using the term “average.” We are required to be [particularly careful with the term “average” as it means different things to different individuals in different aspects. One of the most common uses of the term “average” is what statisticians and mathematicians call the arithmetic mean, or simply mean.

Q2. Which is the Best Method of Finding a Central Value?

Answer: Selecting the most appropriate measure of central tendency typically depends upon the type of dataset you have been given.

Q3. Why is it Called a Measure of Central Tendencies?

Answer: In many distributions, there are values that are more likely and less likely to occur. A measure of central tendency typically determines where values are more likely to occur–or where they *tend* to occur. therefore, “tendency”.


The measure of central tendency is more applicable to the mean and median. Both of these measures identify a central point in the distribution. This central point is where the values are possible to occur.


As we saw with categorical data, there is no central value. Consequently, central doesn’t really apply for the mode. However, we still use the terminology.

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