Measures of Central Tendency

The mean (or average) is the most popular and well-known measure of central tendency. It can be used with both discrete and continuous data, although its use is most often with continuous data. The mean is equal to the sum of all the values in the data set divided by the number of values in the data set. 

You may have noticed that the above formula refers to the sample mean. So, why have we called it a sample mean? This is because, in statistics, samples and populations have very different meanings and these differences are very important, even if, in the case of the mean, they are calculated in the same way. To acknowledge that we are calculating the population mean and not the sample mean, we use the Greek lower-case letter mu, denoted as µ: 

Using the formula of the mean, we have:

Substitute all the values of x, in our case, we have: 

Obviously, the process of getting the mean is correct. However if we look at the computed value, it seems that the 30.7k might not be the best way to accurately reflect the typical salary of a worker, as most workers have salaries ranges from 12k to 18k. The mean is being skewed by the two large salaries. Therefore, in this situation, we would like to have a better measure of central tendency. 

The median is the middle score for a set of data that has been arranged in order of magnitude. The median is less affected by outliers and skewed data. 

Using the formula of the median, we have:

We first need to rearrange that data into order of magnitude (smallest first). Then we have:

After arranging the data, locate the middle score to solve for the median. (Remember: if the number of data is even, locate the two middle score and get their average) 

Since n = 10 (even number), then we locate the two middle score in our data set, we have: 

Comparing the two averages, mean = 30.7k and median=15.5k. The computed average using the median can accurately describe the typical salary of a worker, since most workers have salaries ranges from 12k to 18k. Remember that in the case where outliers are significant in the data set, the median may be the best measure of central tendency. 

The mode is the most frequent score in our data set. On a histogram it represents the highest bar in a bar chart or histogram. You can, therefore, sometimes consider the mode as being the most popular option. 

Normally, the mode is used for categorical data where we wish to know which is the most common category, as illustrated below:

We can see above that the most common form of transport, in this particular data set, is the bus. 

However, one of the problems with the mode is that it is not unique, so it leaves us with problems when we have two or more values that share the highest frequency, such as below: . 

Using the formula of the median, we have:

We first need to rearrange that data into order of magnitude (smallest first). Then we have:

Looking at the data above, there is no frequent score that appears in the data set. Therefore, there is no mode at all. 

(Note: you can’t put a ”0” to represent ”no mode” since ”0” means something) 

To solve for the Mean involving grouped data, we use: 

Steps in Solving the mean involving grouped data 

First: Locate the midpoint m, in the given data.

To find the midpoint, add the class limits and divide it by 2. 

Example: (10 + 12)/2 = 22/2 = 11

so, we have:

Second: Once you have located the midpoint for each group, multiply it with its corresponding frequencies to solve for fm. Then find the summation of all the values under the column frequency (f) and fm (frequency*midpoint)

For example: 4 * 11 = 44

So we have:

Third: Solve for the mean by substituting all the values generated in step 2. 

So we have,

So, the average orders received each day is approximately 17. 

To solve for the Median involving grouped data, we use: 

Steps in calculating for the median involving grouped data: 

a]. Create a cumulative frequency in your table. The cumulative frequency is calculated by adding each frequency from a frequency distribution table to the sum of its predecessors. The last value will always be equal to the total for all observations, since all frequencies will already have been added to the previous total; and 

b]. Locate the median class. In locating the median class, calculate median class = n/2

c]. After solving for the median class, locate its value in the cumulative frequency column, in our case, n/2 = 50/2 = 25, the median class (25) is located in the third row under the cumulative frequency. 

Note: To find the median class, we have to find the cumulative frequencies of all the classes and n/2. After that, locate the class whose cumulative frequency is greater than (nearest to) n/2. 

d]. After locating the median class, create another column in your table for the lower limit, Lm 

e]. Once the table is complete, locate the values of the following: (Refer to your median class in locating the following values) 

• Median Class

• <CF (cumulative frequency)

• Lower Limit (Lm)

• Frequency (f)

• Class interval (i)

In our case, 

• The median class is n/2 = 50/2 = 25

• The <CF is the cumulative frequency before the median class, in our case, the median class' cumulative frequency is 36, and 16 is the number before it. So, the <CF = 16.  

• Lower limit of the median class, is 15.5

• Frequency (f) of the median class is 20, and the

• Class Interval (i) is 3. To solve the class interval (i), Find the highest value (21) and the lowest value (10) in the given class interval and divide it with the number of groups (4). In our case, (21-10)/4 = 2.75 or approximately equal to 3.

Once you already have the following values, substitute it to the formula and do the operation: 

So, the median of the given data set is 17. 

To solve for the Mode involving grouped data, we use: 

Steps in calculating for the mode involving grouped data: 

To solve the Mode for Grouped data, one must locate the group with the highest frequency since mode is define as the most frequent score in the data set. 

In our example, Number of orders, 16 -18 has the highest frequency, f (20). Therefore, the modal class is in the third row under the frequency. 

So we have,

Once the you have located the modal class, identify the values of the following:(Refer to your modal class in finding the following values) 

Once you already have the following values, substitute it to the formula and do the operation.

So, the mode for the given data set is approximately equal to 17.