Assignment

A language teacher obtained the following scores after giving the students an English test.

5 9 7 7 9 8 5 8 9 9 9 7 6 6 7 6 7 8 7 9

Please compute the mean and interpret the results (what does the mean mean?).

Variability

Variability shows how scores vary in a distribution. There are several ways of estimating the variability of scores, but as a language teacher you need to know only two of them: range and standard deviation.

Range
It simply means the difference between the highest score and the lowest score. In the formula, you have to add 1 because you have to take into account the upper limit of the highest score (0.5) and the lower limit of the lowest score (0.5).

Range = H - L + 1

Standard deviation
Standard deviation shows how far a particular score deviates (moves away) from the mean. You can compute standard deviation by using the following formula:


The smaller the standard deviation, the more homogeneous the group of scores is. The larger the standard deviation, the more heterogeneous it is.

Central Tendency

There are three methods of measuring central tendency. I believe you are familiar with all of them as you learned about them in primary school.

Mode
Mode simply means the number which occurs most frequently. For example, you have the following scores:

9 9 8 8 7 7 7 6 6 5 5

The mode is 7 because this score occurs three times, while the others only occur twice.

Median
Median is the number which is located exactly in the middle of the distribution. Before determining the median of the distribution, make sure you list the scores from the highest to the lowest. If you have odd number of scores in a distribution, pick the score in the middle. For example:

9 9 8 8 7 7 7 6 6 5 5

There 11 scores in the above distribution. The median is 7.
However, if there are even number of scores in the distribution, pick two scores in the middle, add them and divide the result by 2.

9 9 8 8 8 7 6 6 5 5

There are 10 scores in the above distribution. Pick 8 and 7 which are located in the middle of the distribution. (8 + 7) / 2 = 7.5. The median is 7.5.

Mean
Mean is the average of the scores in a distribution. You compute the mean by dividing the total sum of scores by the number of scores. The formula is:

Types of Scales

The term scales here simply means numbers. Before you decide the formula you're going to use in statistical analysis, you have to know what kind of numeric data you have. Different type of numbers (numeric data) needs a different formula.

There are 4 types of scales.

Nominal scale
It is used only to label or name something, and can be replaced with letters. It has no numeric value so it cannot be added, subtracted, multiplied or divided. For example:


The above question results in numeric data in the form of nominal scale. If the respondent is male, he will answer 1. If the respondent is female, she will answer 2. The numbers 1 and 2 are used to label the respondents on the basis of their gender. We can use options [a] for male and [b] for female instead of numbers.

Ordinal scale
It is used to show ranks. It has no numeric value so it cannot be added, subtracted, multiplied or divided. For example, the following ranks (the blue column) in Class 1A at Mbebekan Elementary School in semester 1 2009/2010 show ordinal scale:


Interval scale
It is used to show numbers which have equal interval between one unit and another. It can be added, subtracted, multiplied or divided. However, it has no absolute zero. The examples are numbers showing temperature and scores.

Ratio scale
Like interval scale, it is used to show numbers which have equal interval between one unit and another. It can be added, subtracted, multiplied or divided. It has absolute zero, so the number 0 indicates nothing or no one. The examples include numbers showing length, height, weight or time.