Introduction
A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. As such, measures of central tendency are sometimes called measures of central location. They are also classed as summary statistics. The mean (often called the average) is most likely the measure of central tendency that you are most familiar with, but there are others, such as the median and the mode.
The mean, median and mode are all valid measures of central tendency, but under different conditions, some measures of central tendency become more appropriate to use than others. In the following sections, we will look at the mean, mode and median, and learn how to calculate them and under what conditions they are most appropriate to be used.
Task
The lesson aims to:
- Identify the measures of central tendency
- Solve the range, mean, median, and mode
- Describe the data in terms of mean, median, and mode.
Process
- Discussion
There are three measures of central tendency, first one is the mean. Mean – is the most commonly used measure of central tendency. It is used to describe a set of data where the measures cluster or concentrate at a point. It is found by adding the values of the data and dividing by the total number of values. It is given with the formula 𝑥̅ = Ʃ𝑥 𝑁 Where: Ʃx = the summation of x or (the sum of the measure) N = number of the values of x First example: 80, 80, 83, 84, 85, 87, 88, 89, 90, 94 So from the formula given, let us now try to analyze and then substitute the given values to the given formula. 𝑥̅ = Ʃ𝑥 𝑁 𝑥̅ = 80+80+83+84+85+87+88+89+90+94 . Hence, the mean grade of the ten students is 86. Second example: The five players of basketball team have the scores of 10, 15, 20, 10, and 25. Find the mean: 𝑥̅ = Ʃ𝑥 𝑁 𝑥̅ = 10+15+20+10+25 5 𝑥̅ = 80 5 𝑥̅ = 16 𝑥̅= 86
Therefore, the mean score of five players of basketball is equal to 16.
Do you understand class on how to find the mean of the given data? Is there any question? Okay let us proceed to the second measure of central tendency.
Median – is the midpoint of the array. The median will be either a specific value or will fall between two values. First example: Using the same data above. The math grades of ten students are 85, 80, 88, 83, 87, 89, 84, 80, 94, and 90. Find the median. Solution: To find the median, we need to arrange first the data in increasing order that is from least to greatest or vice versa. 80, 80, 83, 84, 85, 87, 88, 89, 90, 94 Since the middle point falls halfway between 85 and 87, so in order to get the median, we need get the mean of these two values. Median = 85+87 2 Median = 172 2 Median = 86 Therefore, the median of the given data is 86.
Mode is the most frequent value of a set of data
Select the measure that appears most often in the set. If two or more measures appear the same number of times, then each of these values is a mode. If every measure appears the same number of times, then the set of data has no mode. First example: Using the same data above. 80,80,83,84,85,87,88,89,90,94 Since 80 is the value that most often occur in the given data, therefore 80 is the mode.
Measures of central tendency or averages give us one value for the distribution and this value represents the entire distribution. In this way averages convert a group of figures into one value.
Collected and classified figures are vast. To condense these figures we use average. Average converts the whole set of figures into just one figure and thus helps in condensation.
To make comparisons of two or more than two distributions, we have to find the representative values of these distributions. These representative values are found with the help of measures of the central tendency.
Many techniques of statistical analysis like Measures of Dispersion, Measures of Skewness, Measures of Correlation, and Index Numbers are based on measures of central tendency. That is why; measures of central tendency are also called as measures of the first order.
Seeing this importance of averages in statistics, Prof. Bowley said "Statistics may rightly be called as science of averages."
Evaluation
- Evaluation
Find the range, mean , median, and mode of the following data.
- 100,98,78,99,97,67,78,87,88,98
- 89,88,78,76,76,76,75,99,100,98
- 78,77,80,87,80,80,87,90,92,91
- Assignment
Find the range, mean , median, and mode of the grades of 100 students.
100,98,78,99,97,67,78,87,88,98
78,77,80,87,80,80,87,90,92,91
89,88,78,76,76,76,75,99,100,98
89,88,78,76,76,76,75,99,100,98
78,77,80,87,80,80,87,90,92,91
89,88,78,76,76,76,75,99,100,98
89,88,78,76,76,76,75,99,100,98
78,77,80,87,80,80,87,90,92,91
100,98,78,99,97,67,78,87,88,98
100,98,78,99,97,67,78,87,88,98
| Score | |
| 7-10 |
Very Good Provides excellent quality of work
|
| 5-7 |
Good Provides high quality of work |
| 0-4 |
Poor Provides work that usually needs to be redone or rechecked by others
|
Conclusion
Mean, median, and mode are three kinds of "averages". There are many "averages" in statistics, but these are, I think, the three most common, and are certainly the three you are most likely to encounter in your pre-statistics courses, if the topic comes up at all.
Measures of central tendency are very useful in Statistics. Their importance is because of the following reasons:
(i) To find representative value:
Measures of central tendency or averages give us one value for the distribution and this value represents the entire distribution. In this way averages convert a group of figures into one value.
(ii) To condense data:
Collected and classified figures are vast. To condense these figures we use average. Average converts the whole set of figures into just one figure and thus helps in condensation.
(iii) To make comparisons:
To make comparisons of two or more than two distributions, we have to find the representative values of these distributions. These representative values are found with the help of measures of the central tendency.
(iv) Helpful in further statistical analysis:
Many techniques of statistical analysis like Measures of Dispersion, Measures of Skewness, Measures of Correlation, and Index Numbers are based on measures of central tendency. That is why; measures of central tendency are also called as measures of the first order.
Seeing this importance of averages in statistics, Prof. Bowley said "Statistics may rightly be called as science of averages."