PEP 6305 Measurement in Health & Physical Education

 

Topic 3: Percentiles & Measures of Central Tendency

Section 3.2

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Measures of Central Tendency

 

n   Central tendency is a way to describe the midpoint of a distribution.

n   Measures of central tendency are used to describe performance or ability for a group of subjects rather than describing any individual subject or the whole range of values.

n   Three measures of central tendency:

¨  Mode: the most frequently observed score. To identify the mode, look at the simple frequency distribution and find the value that occurs most frequently. Works best for nominal-scale data.

¨  Median: the 50th percentile; the score at which half the scores lie above and half below. To identify the median, find the midpoint of the rank order distribution. Works best for ordinal-scale data, or interval-scale or ratio-scale data that is highly skewed.

¨  Mean: the average score. Works best for continuous, symmetrically distributed, interval-scale or ratio-scale data. Many variables in health and kinesiology have these characteristics, so the mean is the most frequently reported measure of central tendency.

n   The symbol for the mean is a bar over the variable symbol (in this case, X): . The mean is calculated by summing all of the scores ( ∑ X ) and dividing by the total number of scores (N):

          

n   So, for the example given on page 55 of the text:

    

n   Using the formula, calculate the mean of the data for which you computed the percentiles above. (Answer)

 

 

Relations Among the Measures of Central Tendency

n   The mode, median, and mean are all approximately the same value for interval or ratio scale data that are normally (or even just symmetrically) distributed.

n   When the data distribution is skewed, these values are not the same.

¨  For positively skewed data, the mean has a higher value than the median, and the median has a higher value than the mode.

¨  For negatively skewed data, the mean has a lower value than the median, and the median has a lower value than the mode.

                    

           Positive skew: Mode < Median < Mean                                      Negative skew: Mean < Median < Mode

 

n   How do you know which measure of central tendency to use?

¨  For continuous data that are symmetrically distributed, use the mean.

¨  For ordinal data or continuous data that are highly skewed, the median may be a better representation of the group because it is not influenced by extreme values.

¨  For nominal data, use the mode or, even better, describe the frequency or proportion of subjects in each category (i.e. the grouped frequency distribution).

 

n   The mean and the median (50th percentile) can be produced using R Commander; they were printed when you did the deciles example.

¨  You can see (I hope) that the Mean = 35.19, and the Median (50%-tile) = 35. You can use Excel to order the data in a column and then look to identify the most common value if you need to find the Mode (the most frequent score).

 

Formative Evaluation

 

n   List the advantages and disadvantages of the mode, median, and mean as measures of central tendency (see text, p. 56-57). Describe the data scale type appropriate for each.

n   Give an example of a variable for which you would compute the mode, median, and mean.

n   How do you decide whether to use the mode, median, or mean to represent central tendency?

 

You have reached the end of Topic 3.

Make sure to work through the Formative Evaluation above and the textbook problems (end of the chapters). (remember how to enter data into R Commander?)

You must complete the review quiz (in the Quizzes folder on the Blackboard course home page) before you can advance to the next topic.