Analyzing the Data:Calculating Mean, Median & Mode

After all of the research has been conducted, the next step in the research process is that of analyzing the data to see if any “correlations” or relationships that exist between any 2 or more elements

In order to make sense of the data, researchers use a variety of mathematical tools to help summarize the information.

The three most common are:

Mode

Median

Mean

Let’s take a look at each……..

1. Mean

Step 1: Find the sum of the numbers.          3+5+7 = 15

Example: To find the mean of 3,5,7.

Formula: 

  Mean = sum of elements / number of elements

Step 2: Calculate the total number.          there are 3 numbers.

Step 3: Finding mean.          15/3 = 5

2. Median

Step 2: Arrange the numbers in ascending order.          1,2,4,5,7

• Median is the middle value of the given numbers or distributionExample: To find the median of 4,5,7,2,1

Step 1: Count the total numbers given.        There are 5 elements or numbers in the distribution.

Step 3: Choose the element in the middle    

Example: For an odd number set (1,2,4,5,7) look for the middle element

4 is the median

Example: For an even number set (1,2,4,5,7,9), look for the two middle elements and take their average

4+5=9/2 = 4.5

4.5 is the median

So….what’s the difference between median and mean?  

Aren’t they the same thing, just looking for the middle number?  

Well, the answer is “no”, they’re different….let’s look at an example:

Let’s look at a Grand River Transit bus. At a given time, five people are riding it…..here is their income distribution:

$ 70,000

$ 60,000

$ 50,000

$ 40,000

$ 30,000

The “mean” income of those riders is $50,000 a year (add up the salaries and divide by the number of salaries).

$70,000 + $60,000 + $50,000 + $40,000 + $30,000

= $250,000 / 5

= $50,000

= $70,000 / $60,000 / $50,000 / $40,000 / $30,000

If we calculate the median income of those riders, it is also $50,000 a year.

• So, you get the same answer….why do both ??

•OK…let’s analyze a different scenario…..

Here is the new income distribution:

$ 50,000,000

$ 60,000

$ 50,000

$ 40,000

$ 30,000

•Joe Blow gets off the bus. Bill Gates gets on.

But the mean income is now somewhere in the neighborhood of $10 million or so.

• The median income of those riders remains

$50,000 a year

$50,000,000 + $60,000 + $50,000 + $40,000 + $30,000

= $50,180,000 / 5

= $10,036,000

$50,000,000 / $60,000 / $50,000 / $40,000 / $30,000

•By using the “mean”, the “average” income of those bus riders is just over 10 million dollars

• In this case, using the “median” rather than the “mean” would provide a much more representative and accurate picture of those bus riders' place in the Kitchener economy

• As a result, we need to calculate both the “mean” and “median” and then use them in context to what we are researching

3. Mode

Example: To find the mode of 11,3,5,11,7,3,11

The mode is the most frequently occurring value in a frequency distribution.

•Step 1: Arrange the numbers in ascending order.          3,3,5,7,11,11,11

Step 2:  In the above distribution:

        Number 11 occurs 3 times,          Number 3 occurs 2 times,

          Number 5 occurs 1 times,          Number 7 occurs 1 times.

So the number with most occurrences is 11 and is the Mode of this distribution.

Note: If all of the elements only occur once, there is no mode.

• As a result, a researcher will use the mean, median and mode to help decipher and interpret the raw data they have collected

• The element(s) that is utilized to interpret the data depends on the context of the research and what interpretation “makes the most sense”

The End !!