4. TABULATION & ANALYSIS OF DATA

HOW TO CALCULATE AND TABULATE STATISTICS

In Statistics, the most important calculations are the mean, mode, median, variance and standard deviation. Statistics begins with a set of numbers which are called the Sample. The set of all possible numbers is called the Populations. Let’s say that we have 20 students in a class with their English Test score scale from 50-100. Then the sample size is 20 and the population is the set of all people who have take or will take the test.

The Table shown below is an example of the tabulation of students’ score.

Student	Score
Student 1	55
Student 2	77
Student 3	90
Student 4	66
Student 5	71
Student 6	67
Student 7	66
Student 8	81
Student 9	66
Student 10	88
Student 11	73
Student 12	66
Student 13	85
Student 14	82
Student 15	66
Student 16	76
Student 17	64
Student 18	66
Student 19	90
Student 20	87

To calculate the mean, you sum up all the numbers in the sample and then divide by the sample size. The sum is 1475. The mean is 1475/20 = 73.75.

CALCULATING THE MODE

The mode is the number that appears the most often in the sample.

To calculate the mode, we count the number of times each rating is made. In the sample above, the number that appears the most often is “66” that appeared for 6 times in the table. Hence, the mode is 66.

But what would happen if we have the following sequence : 66,66,70,77,77

In this case, a conclusion that can be draw is that there is no unique mode. A mode is unique if and only if ONE number is more frequent than all others.

What if the sample size is too big such as > 50 ? It would cost a lot of time and may contain inevitable human mistake in arranging the data. Hence, we can make use of Microsoft Excel in calculating the median for us. Firstly, the score is entered into the Microsoft Excel and let’s use the tabulation above as example. As the result, the data entered would range from row A2 to A21. After that, at the column below the last data insert this formula in order to get the MODE ; =Mode(A2:A21) and press enter.

CALCULATING THE MEDIAN

The median is the value we get when we order all of our scores and then find the one in the middle. But what happens if the sample size is even? In this case, we can add the two middle numbers and then divide by 2. For the tabulation above, the sample size is 20 which is even and hence the two middle numbers need to be divided by 2 and the result is 72.

***BEAR IN MIND THAT ALL THE SCORE MUST ARRANGE IN A ASCENDING ORDER BEFORE CALCULATING THE MEDIAN IN ORDER TO GET THE CORRECT RESULT.***

Similarly to the mode, we can use Microsoft Excel to calculate the median of a set of data. Firstly, the score is entered into the Microsoft Excel and let’s use the tabulation above as example. As the result, the data entered would range from row A2 to A21. After that, at the column below the last data insert this formula in order to get the median ; =Median(A1:A20) and press enter.

CALCULATING VARIANCE

The variance is a measure of the variation of the sample data. The larger the variance, the more random the answers appear. Most people find that standard deviation is a more useful way of measuring variability.

The method for calculating the variance is different depending on whether we are calculating the variance of a population (everyone) or the variance of a sample. In order to calculate the variance of a given sample data, one must follow the steps provided below :

1.     Calculate the mean of the sample data.

2.     Calculate the difference between each value and mean. (Negative values might be obtained and it is perfectly normal)

3.     Multiply the answer in Step 2 to the power of 2 or in mathematical term “Square” it.

4.     Sum up all the values calculated in step 3.

5.     For the sample variance, we divide the sum in step 4 by the (sample size -1).

The formula for the variance is :

 = Variance

X = respective sample value

 = Sum of

X bar = Mean or average

N = Sample size

Similarly, Variance could be calculated in Microsoft Power Point by inserting the appropriate formula. Firstly, the score is entered into the Microsoft Excel and let’s use the tabulation above as example. As the result, the data entered would range from row A2 to A21. After that, at the column below the last data insert this formula in order to get the variance; =Var(A2:A21) and press enter.

CALCULATING THE STANDARD DEVIATION

The standard deviation, like variance, is a measure of the variation of the sample data. The larger the standard deviation, the more random the answers appear. Standard deviation is a more popular way of measuring variation compared to the variance. The method of calculating standard deviation is the same as the variance but with one extra step. The steps of calculating standard deviation are :

1.     Calculate the mean of the sample data.

2.     Calculate the difference between each value and mean. (Negative values might be obtained and it is perfectly normal)

3.     Multiply the answer in Step 2 to the power of 2 or in mathematical term “Square” it.

4.     Sum up all the values calculated in step 3.

5.     For the sample variance, we divide the sum in step 4 by the (sample size -1).

6.     Lastly, we take the square root of the value in step 5.

The formula for calculating standard deviation is:

Again, similarly,standard deviation could be calculated in Microsoft Power Point by inserting the appropriate formula. Firstly, the score is entered into the Microsoft Excel and let’s use the tabulation above as example. As the result, the data entered would range from row A2 to A21. After that, at the column below the last data insert this formula in order to get the Standard Deviation; =STDEV(A2:A21) and press enter.

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