Mean, Median And Mode

When we use the word "average", we could mean the mean, the median or the mode because these are all "averages" or more correctly, measures of central tendency.

Mean

Mean - Add up all the data scores and then divide by the number of scores there are.

Median

Median - Write all the scores from smallest to largest and then choose the score in the middle.

Mode

Mode - This is the most frequent score (or most often score).

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Example One - Odd Number of Raw Data

The population numbers of alligators in 7 national parks is below. Calculate the mean, the median and the mode.
Data - 14, 13, 8, 19, 13, 24, 15

Answer:

Mean	= (14 + 13 + 8 + 19 + 13 + 24 + 15) ÷ 7
	= 106 ÷ 7
	= 15.1 alligators

Ordered Data - 8, 13, 13, 14, 15, 19, 24
Median (middle score in ordered data) = 14 alligators

Mode (most often score) = 13 alligators

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Example Two - Even Number of Raw Data

The number of graffiti convictions for 12 months of monthy data is below. Calculate the mean, the median and the mode.
Data - 14, 25, 23, 15, 16, 16, 11, 23, 12, 13, 24, 7

Answer:

Mean	= (14 + 25 + 23 + 15 + 16 + 16 + 11 + 23 + 12 + 13 + 24 + 7) ÷ 12
	= 199 ÷ 12
	= 16.6 convictions

Ordered Data - 7, 11, 12, 13, 14, 15, 16, 16, 23, 23, 24, 25

Median (middle score in ordered data)	= halfway between 15 and 16
	= (15 + 16) ÷ 2
	= 15.5 convictions

Mode = 16 and 23
There are two modes. The data is bimodal.

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Example Three - Effect of an Outlier on the Mean

An outlier is a score which is considerably different from the rest of the data. (For example, one of your classmates may have over 1000 Facebook "Friends"!!) Since the outlier will skew the data analysis, the outlier is often left out. However, in this example, we will include it to show you the skewing effect of the outlier on the mean.

The number of Facebook "Friends" is below. Calculate the mean, the median and the mode.
Data - 12, 15, 3, 10, 25, 1000, 12

Answer:

Mean	= (12 + 15 + 3 + 10 + 25 + 1000 + 12) ÷ 7
	= 1077 ÷ 7
	= 153.9 Friends!!

Ordered Data - 3, 10, 12, 12, 15, 25, 1000
Median = 12 Friends

Mode = 12 Friends

Using Organized Data

In statistics, the data is usually so large that it must be organized into tables with extra columns. The rules to calculate the mean and the median are adapted to suit this tabular format.

Mean of Organized Data
Mean =	∑ fx	(where Sigma ∑ means "the sum of")
	∑ f

Median of Organized Data
The median score is halfway through the data scores so the median is identified using the rule:

( ∑ f + 1 )

2

where
∑ f is the the sum of the frequencies. Refer to the examples below.

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Example Four - Organized Data

Calculate the mean, the median and the mode from this road accident fatalities data:

AGE (YEARS)	NUMBER OF DEATHS
16	5
17	8
18	20
19	27

Answer:

AGE (YEARS) x	NUMBER OF DEATHS f	f x
16	5	16 × 5 = 80
17	8	17 × 8 = 136
18	25	18 × 25 = 450
19	27	19 × 27 = 513
	∑ f = 65	∑ fx = 1179

Mean =	∑ fx	=	1179	= 18.14 years
	∑ f		65

Median score =	(∑ f + 1)	=	(65 + 1)	= 33rd score = 18 years
	2		2

(The data has already been ordered from smallest to largest in the table.
Count down the frequency column to find the 33rd score.
The 33rd person is in the row for the 18 years' age group.)

Mode = 19 years

Example Five - Real Estate Prices

Real estate agents and property developers only use the median house price.
However, in this example of house prices, we will work out the mean, the median and the mode.

HOUSE PRICES ($)	NUMBER OF HOUSES
100 000 - 199 999	10
200 000 - 299 999	12
300 000 - 399 999	13

Answer:

HOUSE PRICES ($) CLASS	HOUSE PRICES ($) MIDPOINT x	NUMBER OF HOUSES f	f x
100 000 - 199 999	149 999.50	10	1 499 995.00
200 000 - 299 999	249 999.50	12	2 999 994.50
300 000 - 399 999	349 999.50	13	4 549 993.50
		∑ f = 35	∑ fx = 9 049 983.00

Mean =	∑ fx
	∑ f
=	9 049 983.00	= $ 258 570.94
	35

Median score =	(∑ f + 1)
	2
=	(35 + 1)	= 18th house price
	2
		= $200 000 to $299 999

(The data has already been ordered from smallest to largest in the table.
Count down the frequency column to find the 18th score.
The 18th house price is in the 2nd row.)

Mode = $300 000 to $399 999