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Averages
Averages
Calculating Averages
Calculating Averages
Often averages are used to summarise all the values in a set of data.
For example: What would you do to answer the question - "On average for how many hours do you exercise each week?" The person that asked he question isn't looking to be given a bunch of numbers of the last 10 weeks, rather they just want one.
There are three different statistics here we can use the measure an average amount.
These are:
Mean - this is all of the values added together, divided by the number of values there is.
Median - this the middle number when sorted in order
Mode - this is the most common number to occur
Example: Find the mean of 5, 7, 2, 4, 9, 4, 6
Example: Find the mean of 5, 7, 2, 4, 9, 4, 6
Add all the values
5+7+2+4+4+6 = 28
Divide the total by the number of values
In this case there are 7 values so 28 ÷ 7 = 4
The mean is 4
Example: find the mode of 5, 7, 2, 4, 9, 4, 6
Example: find the mode of 5, 7, 2, 4, 9, 4, 6
For this we look for the most common value which in this case is 4
So the mode is 4
Sometimes there can be two modes or no mode!
Example: find the median of 5, 7, 2, 4, 9, 4, 6
Example: find the median of 5, 7, 2, 4, 9, 4, 6
First we need to sort the values in order
2, 4, 4, 5, 6, 7, 9
Next we cross one off each side until we reach the middle
2,4,4, 5,6,7,9
This leaves 5. Therefore the median is 5.
However if there are two numbers left we need to take the mean of these numbers
2,4,4, 5, 6,6,7,95+6=11
11 ÷ 2 = 5.5 (So in this case the mean is 5.5)
Which Average Do I Use?
Which Average Do I Use?
So we have three different ways of finding an average. How do we know what one to choose?
Well for each of them they definitely have there positives and negatives.
Mode
Mode
+ Really easily to find
- Does not use all of the data
Mean
Mean
+ Fairly easy to calculate, uses all of the data to produce value
- Is affected by extreme values/outliers
Median
Median
+ Uses all of the data to find where the middle value lies, and is not affected much my extreme values
- Can be time consuming to sort the values, and at times easy to make a mistake
Therefore if your data has extreme value/s (numbers that are much smaller or larger than the rest) it is best to use the median.
Therefore if your data has extreme value/s (numbers that are much smaller or larger than the rest) it is best to use the median.
Quartiles
Quartiles
Quartiles are numbers that split the data into quarters. To split the data into four parts there must be three divisions:
The median - half the data above that value and the other half below
The Lower Quartile (LQ) - this is the median of the lower half
The Upper Quartile (UQ) - this is the median of the upper half
For the above data:
The LQ is 3, the median is 5.5, and the UQ is 7
Spread
Spread
Spread is how we measure how apart the data is. There are two measures of spread:
Range
Range
Range= Maximum - Minimum
Interquartile range (IQR)
Interquartile range (IQR)
Interquartile Range = Upper Quartile - Lower Quartile
Out of the two methods the IQR is more reliable as it is less effected by extreme values. Whereas one extreme value could make the range really large when actually the rest of the values are grouped closely to one another.
For each of the questions in the above practice questions for quartiles - find the IQR and Range.
For each of the questions in the above practice questions for quartiles - find the IQR and Range.
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