Chapter 13: Univariate data
13A: Measures of central tendency
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Measures of central tendency are methods that we use to look at the middle or centre point of the data
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There are three different ways of doing this:
_____________: the average of all observations in a set of data (𝑥̅ )
______________: the middle observation in an set of data that is put in order
___________: the most frequent/common observation in a data set
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Grouped vs ungrouped data sets
Ungrouped means each individual data observation is looked at within the data set
Grouped means that the data has been put into different groups or intervals, rather than looking at each
data observation separately
•
means ‘sum’ or ‘total’
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is called ‘x bar’ and is the symbol for ‘mean’ or ‘average’
Calculating the mean, median and mode using UNGROUPED DATA (13A part one)
MEAN
To find the mean (average) of the data set:
1.
all the observations/scores in the data set together (they do not have to be in order)
2.
by the number of observations/scores
We can write this as:
Or, as:
where x is the scores
and n is the number of scores
Worked example
Find the mean of the data set: 6, 2, 4, 3, 4, 5, 4, 5
1. Add the observations/scores together (in other words, find ∑ 𝑥 which is the total/sum of the scores)
∑𝑥 = 6 + 2 + 4 + 3 + 4 + 5 + 4 + 5
∑𝑥 =
2. Divide by the number of scores (n)
There are 8 scores in this data set (n = 8)
𝑥̅ =
𝑥̅ =
𝑥̅ =
∑𝑥
𝑛
MEDIAN
To find the median (middle/centre score) of the data set:
1. Arrange the scores in numerical order (smallest to biggest is the easiest way)
2. Put one finger on the smallest score, and a finger on the biggest score, and move your fingers inward one
number at a time until they meet at the middle score
3. If there are an
number of scores, the median is the middle score
4. If there are an
number of scores, find the mean/average (𝑥̅ ) of the two middle scores
Worked example
Find the median of the data set: 6, 2, 4, 3, 4, 5, 4, 5, 3
1. Put the scores in numerical order:
2, 3, 3, 4, 4, 4, 5, 5, 6
2. Working inwards from the smallest and biggest numbers, we find that the middle score is
Therefore, the median of this data set is
MODE
To find the mode (most frequent/common score) of the data set:
1. Work through the data set and record how many times each score appears (it might be easier to put them in
order first to ensure you don’t miss any)
2. Whichever score appears most frequently/commonly is the mode
Note:
• Sometimes there is no mode – each score appears once only
• Sometimes there is one clear mode – one number that appears most frequently/commonly
• Sometimes there is more than one mode
Worked example
Find the mode of the data set: 6, 2, 4, 3, 4, 5, 4, 5, 3
1. (optional, but useful) Put the scores in numerical order:
2, 3, 3, 4, 4, 4, 5, 5, 6
2. Determine which number (or numbers) appear most commonly
In this case, the mode is
(it appears
times in this data set)
Using a FREQUENCY TABLE to calculate mean, median and mode
1. Draw up a table with 4 columns: Score (x), Frequency (f), Frequency x score (fx), Cumulative frequency (cf)
*We find the MEAN using this formula:
f = frequency, x = the scores
*We find the MEDIAN by finding the position of each score in cumulative frequency column
We then use the formula
to find where (at what position) the median will appear, then reading
the score from the cf
*We find the MODE by looking for the score with the highest frequency
Worked example
Score (x)
Frequency (f)
4
1
5
2
6
5
7
4
8
3
n
Total ∑
Frequency x score (fx)
Cumulative frequency (cf)
∑ (fx)
(not needed)
What do you do in the ‘frequency x score’ column?
What do you do in the ‘cumulative frequency’ column?
Then, use the data to calculate the mean, median and mode.
MEAN
Use the formula:
𝑥̅ =
𝑥̅ =
MEDIAN
Locate the position of the median using
Median position =
=
Use the
2
, which means that the median is the 8th score
column to find the 8th score, which is
MODE
The score with the highest frequency is
, therefore
is the mode