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Measures of location

There is more than one measure of the "typical" value in a list, these being: mean, median, mode.

Mean.  The arithmetic mean is found by adding up all the numbers and dividing by the total number of observations. For our data, the mean is

(2+5+7+1+5+6+12+5)/8 = 5.375

Remark: the other types of mean are: geometric mean and harmonic mean

Mode. This is the number in a list that is most frequent. In our example, the mode is 5 as 5 appears 3 times, which is more than any other number.

Median. This is the number above and below which half of the numbers lie. To find the median, 1st arrange the numbers in ascending order (easy to forget this step), next pick out the middle value from the rearranged list by picking the:

  • (n+1)/2 th value if the total number of observations denoted by "n" is odd
  • the mean of the n/2th and (n/2  + 1)the value observation if n is even.

In our example:

a. rearranged in ascending order we have: 1,2,5,5,5,6,7,12

b. We have eight observations, so n = 8. So we pick out from the list the mean of the 4th and 5th observation in the rearranged list. The 4th highest number is 5. The 5th highst number is 5 also. So the mean of the two is also 5 (=(5+5)/2).

The median is 5.

Quartiles. The median is a specific case of what is called a quartile. In summary stats, you may also report key quartiles, so we'll talk about the basics. The lower quartile is the value below which one quarter of the observations fall, and the upper quartile is the value below which three quarters of the observations fall. Lower and upper quartiles are also known, respectively, as 25the and 75th percentiles. You will find slight differences in how to calculate these figures from one book to the next. The methods come up with about the same answers.The lower and upper quartiles are used to calculate the interquartile range (IQR) that gives you an idea of the spread of the data.(See section on measuring spread.)

Mean, median, or mode?

Given the 3 measures of central location, you might be asking "Do I have to report all 3 numbers?", "Does it matter which one I should report?"

Each measure has its pros and cons, and taking these into account, which measure you use depends on the data at hand.

  • nominal - report mode;
  • ordinal - report median; mean if we code up the ordinal data; (mode is also ok)
  • continuous - report mean; median when data is skewed; (mode is also ok)

Questions: In our example, the median=mode (=5). This is not always the case. Can we make a comparative statement such as "the mean is always greater than the mode and median?" Can there be a case when the 3 measures are the same ie mode=median=mode?

Discuss any issues with descriptive stats in the forum.