Statistics · Ordinary Level
The Three Averages
Ordinary Level · Class 2 · Tap NEXT to begin
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Mode, Median & Mean
The three averages
Mode = most often. Median = middle in order of size (for $n$ numbers, the $\tfrac{n+1}{2}$-th; if $n$ is even, average the two middle ones). Mean $\bar{x} = \dfrac{\text{sum}}{\text{how many}}$.
Worked example — all three
Find the mode, median and mean of $3, 1, 5, 1, 100$.
Mode $= 1$
Order $1,1,3,5,100$ → median $= 3$
Mean $= \tfrac{110}{5} = 22$
mode $1$, median $3$, mean $22$
Worked example — odd number
Find the mode, median and mean of $3, 1, 7, 3, 4, 8, 20$.
Mode $= 3$
Order $1,3,3,4,7,8,20$ → median $= 4$ (the $4$th)
Mean $= \tfrac{46}{7}$
mode $3$, median $4$, mean $\tfrac{46}{7}$
Worked example — even number
Find the mode, median and mean of $3, 5, 2, 7, 9, 12$.
Mode: none
Order $2,3,5,7,9,12$; median $= \tfrac{5+7}{2} = 6$
Mean $= \tfrac{38}{6} = 6\tfrac{1}{3}$
none, $6$, $6\tfrac{1}{3}$
Worked example — just the mean
Find $\bar{x}$ of $3, 1, 5, 7, 400$.
$\bar{x} = \tfrac{3+1+5+7+400}{5} = \tfrac{416}{5}$
$83$
Outliers
An outlier is a value way bigger or smaller than the rest — it drags the mean, so the median is often fairer.
Worked example — with an outlier
Find the mode, median and mean of $3, 5, 1, 7, 2, 3, 150$.
Mode $= 3$
Order $1,2,3,3,5,7,150$ → median $= 3$
Mean $= \tfrac{171}{7} = 24.4$ — pulled up by the outlier $150$
mode $3$, median $3$, mean $24.4$
You try
Find the mode, median and mean of $3, 5, 2, 9, 4, 8, 11, 15$.
Order them; $n=8$ (even) so average the two middle values.
Mode: none
Order $2,3,4,5,8,9,11,15$ → median $= \tfrac{5+8}{2} = 6.5$
Mean $= \tfrac{57}{8} = 7.1$
none, $6.5$, $7.1$
Advantages & disadvantages
Mode: easy to find, but may be none. Median: unaffected by outliers, but not useful in other statistics. Mean: very useful in other statistics, but distorted by outliers.
That’s Class 2.
Mode, median and mean, outliers, and the advantages of each. Class 3: working with the mean and the spread.