Measures of central tendency examples with Solutions
Measures of central tendency examples with Solutions

This article "Measures of central tendency examples with Solutions" will be very helpful to understand the idea of measures of central tendency notes and best measure of central tendency.

Measures of central tendency examples with Solutions

A measure of central tendency can be described as a value describing how much to which a variable can vary, over time or over all data points in the data set. A central tendency value tells us what is most likely going to happen in future values. Also called “Central Limit Theorem”.


The measure of central tendency is measured in percentage change, where percentage change is used instead of absolute change. It can also be expressed as a number, if such value is not positive then it is positive, otherwise — negative.


Measures of central tendency can be found in many types of statistics and are widely applied when dealing with variables. The main aim of this post is to show you how they operate by providing two measures of central tendency example.


Measures of Central Tendency:


a. Normal Mean


Mean = Median + std. Mean (Stochastic)


b. Std. Deviation from the mean


Std. Error of the mean = Standard deviation of the data


We have already established that we can use either one or both at once. We will not discuss them separately. Let’s start with the first one. This is a simple measure of central tendency. The normal mean is a mathematical model for measuring the average value of any variable on its own, based on a single data point. If the mean of some variable in a data set occurs in every data points, so mean value should be equal to total amount of times value occurrence in all data points.


Normal Mean


A standard normal distribution has been chosen in order to represent the expected values of mean. This means that the expected value is 0, so we can consider it equal to total amount of time value occurs in all data points. In this equation we use an index 0 and assume that each data point is normally distributed, meaning, data should have an equal chance of occurring.


Mean = Total Amount of Time Value Occurred In All Data Points


Mean = 0.5 * 100 = 0


The value of mean is what we’re trying to predict or find by doing tests or analyzing statistical data. More generally, we want to find the value, where probability of occurrence of value equals 0.1, i.e., 

P(X, Mean) = 0.1.


In order to do this, you need to calculate frequency of occurrence of value by taking the sum and dividing by total amount of time. For example, if value occurred in 50% of all data points, then it should be 

0.5 * 50 = 0.5. 

Now sum up occurrence number for this particular value and divide it by number of data points.


Average Amount Of Time (Mean) in N Values


Mean = 0.5 * 50 = 0


If value occurred in 10 times more times then than normal average, then it should be less — 10 = 0.5. And so on.


To answer this question we will build a formula for calculating frequency —


50 = (Value - Normal Average)/(Values — Normal Average)


50 = (0.1 — 0.1) / Number of occurrences


50 = [0.1 — 0.1]/[2] = 0.5


The formula is easy to follow and remember. If value appeared in 50% of all data points, then it should be 0.5 * 50 = 0.5. As you can see, the value of probability of occurrence equals 0.


But there are still a few problems when interpreting frequency as percentage. First, when we don’t know exactly what value occur which specific thing, then frequency of occurrence should be calculated by drawing random samples. There is no guarantee that what value occurs exactly in a given sample will occur in a corresponding sample. So, to make sure that value was really occurring, we should know exactly what happened with it before, but there is no way to know about what values were changed and whether they were real or fake. So, frequency of occurrence should be estimated using samples. We should also estimate frequency for values of value that did not occur in any sample.


There are several approaches to estimation of frequency. Generally, the simplest approach is taking a sample from each value and estimating frequency. Another one is a weighted sample. However, to estimate the frequency of occurrence of value in general population, we need to estimate frequency of all values of value and divide these values by their size. Therefore, frequency can easily be calculated by using sample of whole values, as well as frequency for different values, then we have to sum and divide that sum by the total amount of time value. For example:


(Value - Normal Average)/(Values — Normal Average) = 50


50 = (0.1 — 0.1) / Number of objects


50 = (0.1 — 50) / Number of objects


50 = (0.1 — 100) / Number of objects


50 = (0.1 — 1000) / Number of objects


50 = [0.1 — 0.1] / Number of Objects


50 = (0.1 — 0.1) / 3 = 0.2


Now, to get the value of frequency, take frequency of occurrence of value and multiply a given number by a given value that occur in all values.


For example, in case of 10 objects in test and 100 in population, then frequency is equal to 0.2 * 100 = 0.2. Frequency of occurrence is equal to frequency of occurrence of value divided by the ratio of values — 10/100. As we have shown above, value that appear in 1% of all objects (value = 0.1) should occur only once. So, value that appeared in 1% of all objects (value = 0) should occur twice as often as value that appeared in 10 objects (value = 0.2). Thus, value that appears in 1% of all objects (value = 0) should occur in 20 times as many times in entire population as value that appeared in 100 objects (value = 0.2).


So, to calculate frequency we should divide the sum, by sample size, by the ratio of occurrence, to sample size which will give us the following formula.


Frequency = Value / Sample Size / Sample Ratio


Frequency = 2/100 = 0.2


Now we should estimate frequency by taking one more sample from each value and multiplying frequencies by sample size. Then we can update frequency by dividing frequencies by sample size, to sum up frequency, divide it by 100 and sum it up again.


Frequency = 2/100 = 0.2


So, value that appear in 1% of all objects (value = 0) should occur 200 time more in entire population. This means in 1% of entire population, value that appeared in 1% of all objects should occur 200 times more than value that appeared in 10 objects 

(value = 0.2).