Central tendency
Frequency tells us how many times an observation occurs in a statistic. In the table below, x shows the grade, f is the frequency, i.e., how many of those grades there are, and f% is the relative frequency, i.e., the relative proportion of grades.
It can be seen from the table that, for example, a grade of 8 is given ten times and that is 20% of all grades.
Example 1
Below are the grades for all of Liisa-Petter's high school courses.
7,7,5,4,6,7,8,7,4,5,8,9,8,7,8,6,5,4,5,6,7,8,6,4,6, 7,8,9,5,5,4,6,7,8,9,9,8,9,10,9,7,8,8,8
Let's make a frequency table.
The centre of the data can be described by central tendency. Which of these methods best describes the centre of the material and what is appropriate for the current purpose depends on what kind of centre information is desired from the material and, of course, it depends on the material itself.
Here we look at three central tendencies: the arithmetic mean, the median, and the mode.
The arithmetic mean
The arithmetic mean is the sum of all measurements divided by the number of observations in the data set.
Example 2
Find the mean of the numbers 3,3,4,7,7,9
Example 3
Find the mean of the grades in Example 1
Because there are a lot of grades we can use the frequency to our advantage. We know that there are five times grade 4 and 4 + 4 + 4 + 4 + 4 = 5 ∙ 4
The mean of all Liisa-Petter's grades is 6.8
The median and the mode
The median is the middle value of the material that is set in the order from smallest to highest. If there are an even number of numbers, the median is the average of the middle two numbers.
Example 4.
1,3,4,7,9
There are five numbers and they are in order. The median is 4.
3,3,6,8,10,11
There are an even number of numbers, ie 6 numbers. The middle numbers are 6 and 8, so the median is the mean of these, that is 7.
Example 5
Find the median of the grades in Example 1
The median is in the middle of the material, so 50% of the grades are left on both sides of it. It can be seen from the frequency table that 50% is met at grade 7
There are an even number of grades and the middle grades are 22 and 23. Both are grade 7
The median is 7
Mode tells us the most common value of the material. That is, the one with the highest frequency. In the above material, the maximum frequency is with grade 8, there are 11 of them.
So the mode is 8.
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