## 8.3: Measures of Central Tendency

Based on Section 8.3 in Finite Mathematics and Finite Mathematics and Applied Calculus

### Mean, Median, and Mode of a Sample

Consider the following scenario: you are interested in the length of time a typical cell phone lasts, so you ask 10 randomly chosen cell phone users how long their last phone lasted. These 10 people are referred to as your sample. Ideally, you should instead poll every single cell phone user, or the population, but that would be virtually impossible and certainly impractical. Statisticians study what we can infer about the whole population from a sample.

Going back to the poll of cell phone users, you find the following figures (in months):

0, 4, 10, 20, 25, 8, 4, 5, 1, 18

Q I have just purchased a cell phone. Based on the above data, how long can I expect it to last?
A That depends on what, exactly, you mean by "expect it to last". In a sense, you are asking for a measurement, or a statistic, to estimate where there "center" of the data lies. Perhaps the most commonly used statistic is the average, or mean, which is computed by adding the scores and dividing the sum by the number of scores in the sample, called the sample size:

 Sample Mean = x = 0 + 4 + 10 + 20 + 25 + 8 + 4 + 5 + 1 + 1810 = 9510 = 9.5

Q Why is the sample mean called x?
A Underlying this calculation is a random variable X, the number of months a cell phone will last. The values 0, 4, 10, ... above are measured values of X, sometimes called the x-scores. The bar over the x indicates that we are taking the average of the set of x-scores.
One more thing: if, instead of the mean of a sample, we have the mean of the entire population, we refer to it as and not x:

x = sample mean
= population mean
Both are compted in exactly the same way.

 The mean of the sample 2, 5, 8, 10 is x = The mean of the population 0, 0, 5, 15, 100 is =

Q OK. The mean is the single statistic tells you the middle of a sample of data, right?
A Take a look at the scores 0, 0, 5, 15, 100 above, and pretend they were the scores received by members of your study group in an English exam. Although the average is 24, it does not reflect the "typical" scores, which were very low. Here are two more statistics we can use:

• The middle score is 5: Two people got more than 5 and two people got less. The middle score in a sample or population called the median.
• The most common score was 0, since more people got a 0 than any other score. A score occurring the most frequently in a sample or population is callsed a mode. (There can be several modes in a set of data.)
Median and Mode

The median m is the middle score (in the case of an odd-size sample or population), or average of the two middle scores (in the case of an even-size sample or population) when the scores are arranged in ascending order. A mode is a score that appears most often in the sample or population. (There may be more than one mode in a sample or population.) As before, we refer to the population median and population mode if the sample consists of the data from the entire population.

Examples To get the median of 2, -2, -2, 0, 2, 1:

First arrange the scores in ascending order: -2, -2, 0, 1, 2, 2
Since there is an even number of scores, select the middle two and take the average: (0 + 1)/2 = 0.5
Median = 0.5
There are two modes: -2 and 2 because those are the scores that occur most often.
Modes: -2, 2

Here are some for you. If enterinmg more than one mode, use commas to sperate them. For instance, if the modes of a sample are 3 and 4, enter the answer as 3,4

 Sample: 0, 2, 4, 4, 10 x = median = mode(s) = Sample: 1, 1, 2, 3 x = median = mode(s) = Sample: 1, 1, 0, 0, -1, 2 x = median = mode(s) =

### Expected Value of a Random Variable

Now, instead of looking at a sample of values of a given random variable, we can look at the probability distribution of the random variable and predict the sample mean without taking any samples. This prediction is what we call the expected value of the random variable.

Expected Value of a Random Variable

If X is a finite random variable that takes on the values x1, x2, ... , xn, then the expected value of X, written E(X) or , is given by the formula

= x1.P(X = x1) + x2.P(X = x2) + ... + xn.P(X = xn)
Multiply each value of X by its probability and add the results.
Interpretation The expected value of X is a prediction of the mean of a large random sample of measurements of X. in other words, it is what we "expect" the mean of a large number of scores to be.

Example Suppose X has the distribution shown:

 x 0 1 2 3 4 P(X = x) 0.1 0.2 0.4 0.2 0.1

Then we can get as shown:

 x 0 1 2 3 4 P(X = x) .1 .2 .4 .2 .1 x.P(X = x) 0.1 = 0 1.2 = .2 2.4 = .8 3.2 = .6 4.1 = .4

So, = Sum of numbers in the third row = 0 + .2 + .8 + .6 + .4 = 2.

Here is one for you to do. Distribution of X:

 x 50 150 250 350 450 P(X = x) .1 .3 .2 .4 0 x.P(X = x)

 =

Here is an application similar to those you can find among the review exercises for this chapter.

The soccer team at Union State U has not been doing so well with referees. The team's players are issued cautions (yellow cards) or sent off the field (red cards) at an alarming rate as evidenced in the following table, which shows the number of games last season during which the team was issued no cards (no games), the number of games during which it was issued 1 card (2 games) , the number of games during which it was issued 2 cards (3 games), and so on:

 Cards 0 1 2 3 4 5 Number of Games 0 2 3 3 6 6

We would like to compute the expected number of cards in a typical game. To do this, we should take our random variable X to be:

 The number of games The number of cards per season The average number of games The number of games per card The number of cards in a game The number of games per season

Now complete the following table, based on the correct choice of X above:

 x 0 1 2 3 4 5 P(X = x) x.P(X = x)
 Therefore, the team can expect to average cards per game.

Now try some of the exercises in Section 8.3 of Finite Mathematics and Finite Mathematics and Applied Calculus

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Last Updated: November, 2003