5. Discuss in detail the measures of central tendency with suitable example.Central tendency refers to the descriptive statistic that best represents the center of a data set, the particular value that all the other data seem to be gathering around. The key measures of central tendency are:
Mean, the Arithmetic AverageThe most commonly reported measure of central tendency is the mean, the arithmetic average of a group of scores. It is calculated by summing all the scores in a data set and then dividing this sum by the total number of scores.
For example, if we explore the numbers of top finishes that countries had in World Cup soccer tournaments, the mean would be calculated by first adding the number of top finishes for each country, then dividing by the total number of countries. For the 14 countries that had at least 1 top finish:
4 + 8 + 1 + 2 + 1 + 2 + 2 + 6 + 2 + 2 + 2 + 2 + 2 + 10 = 46
In this case, we divide 46, the sum of all scores, by 14, the number of scores in this sample:
46/14 = 3.29
Visual Representations: Mean is the visual point that perfectly balances two sides of a distribution.
Formula: Mean = Sum of the values/Number of values
Median, the Middle ScoreThe second most common measure of central tendency is the median. It is the middle score of all the scores in a sample when the scores are arranged in ascending order. We can think of the median as the 50th percentile.
Example: the top finishes for 13 of the 14 countries in the World Cup example are:
4, 8, 1, 2, 1, 2, 2, 6, 2, 2, 2, 2, 10
To determine the median, follow these steps:
Step 1: Arrange the scores in ascending order:
1, 1, 2, 2, 2, 2, 2, 2, 2, 4, 6, 8, 10
Step 2: Find the middle score (With an even number of scores, there will be no actual middle score. In this case, take the mean of the two middle scores.)
There are 13 scores: 13/2 = 6.5. If we add 0.5 to this result, we get 7. Therefore, the median is the 7th score. We now count across to the 7th score. The median is 2.
Formula: Median = (n+1)/2 ranked value
Mode, the Most Common ScoreThe mode is the most common score of all the scores in a sample. It is readily picked out on a frequency table, histogram, or frequency polygon.
Example: Determine the mode for the World Cup data for the 14 countries. The mode can be found either by searching the list of numbers for the most common score or by constructing a frequency table:
1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 4, 6, 8, 10. Mode =2
When a distribution of scores has one mode, we refer to it as unimodal. When a distribution has two modes, we call it bimodal. When a distribution has more than two modes, we call it multimodal.
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To summarize, the central tendency refers to the “typical” score of a group of scores. Mean, median and mode are the three measures of central tendency most commonly used.
Statistics for the Behavioral Sciences, Susan A. Nolan and Thomas Heinzen
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