Describing Data: Measures of Central Tendency

Describing Data: Measures of Central Tendency

Elizabeth Borres

At the end of the lesson, students will be able to:

• Calculate the arithmetic mean, median, mode, and weighted mean.

• Explain the characteristics, uses, advantages, and disadvantages of each measure of location.

Let us review!!!

(Prepare your paper, pen and calculator)

Multiple Choice

What is another word for mean?

• average

• sum

• middle

• frequent

Multiple Choice

Find the mean of the given data 8, 7, 5, 9, 6, 13.

• 8

• 7.5

• 7.5

• 10

Multiple Choice

Find the mean of these numbers:

5,11, 2, 12, 4, 2

• 4.1

• 6

• 4.5

• 4

Multiple Choice

What is another word for median?

• average

• sum

• middle

• frequent

Multiple Choice

What is mode?

• the number that occurs the most in a data set

• the average of all the numbers

• the middle number in the data

• the range of the data

What is an average?

It is a single number used to describe the central tendency of a set of dat.

Examples of an average?

1. The average length of the school year for students in public schools in the United States is 180 days.

2. The average salary of major league baseball players on opening day 2002 was $2, 384, 236 (baseball.espn.go.com)

3. The median asking price for a group of houses listed for sale by a Toledo realtor is $128,000.

4. White-collar pay, which averaged $19.35 per hour, was the highest among occupational groups.

There are several types of AVERAGES.

MEAN, MEDIAN, MODE

MEASURES OF LOCATION

The purpose of a measure of location is to pinpoint the center of a set of observations.

Arithmetic Mean (mean)

• The sum of observations divide by the total number of observations.

• It is the most widely used measure of location.

Parameter

• It is a characteristic of a population.

• Any measurable characteristics of a population.

• It simply means, the mean value taken from the population.

Statistic

• It is a characteristic of a sample

• Any measure based on sample.

• It simply means, the mean value taken from the sample.

Example:

The Kellogg Company had quarterly earnings per share of

$0.89, $0.77, $1.05, $0.79, and $0.95. What is the average quarterly earnings?

•  $\mu\ =\ \frac{\Sigma X}{N}$  

$0.89, $0.77, $1.05, $0.79, and $0.95. What is the average quarterly earnings?

•  $\mu\ =\ \frac{\Sigma X}{N}$  

•  $\mu\ =\ \frac{\left(\text{0.89+}0.77+\text{1.05+0.79+0.95)}\right)}{5}$ 

•  $\mu=\frac{4.45}{5}=0.89$  

• The mean quarterly earning per share is $0.89.

Multiple Choice

To find the average of a set of numbers, add up all the items and divide by...

• 2

• The minimum

• The maximum

• The number of items

Multiple Choice

The height of three basketball players is 210 cm, 220 cm and 191 cm.

What is their average height?

• 210

• 207

• 220

• 200

Multiple Choice

What is the definition of the mean?

• The average

• The middle number

• The difference between the highest and lowest numbers

• The number that occurs the most

Multiple Choice

It is any measurable characteristic of a population.

• parameter

• statistic

Multiple Choice

It is any measurable characteristic of a sample.

• parameter

• statistic

Properties of the MEAN

• Every set of interval level and ratio level data has a mean.

• All data values are used in the calculation.

• A set of data has only one mean, that is, the mean is unique.

• The mean is useful measure for comparing two or more populations.

• The sum of the deviations of each value from the mean will alwyas be zero, that is:

   $\Sigma\left(X\ -\ \overline{X}\right)=0$  

WEIGHTED MEAN

• It is a special case of the arithmetic mean.

• The value of each observation is multiplied by the number of times it occurs. The sum of these products is divided by the total number of observations to give the weighted mean.

• It is often useful when there are several observations of the same value.

Example

• Suppose the coronary care unit has ten employees: 2 aides who earn $12 per hour, 3 nurses' assistants who earn $15 per hour, and 5 registered nurses who earn $21 per hour. Find the weighted average.

Properties of the mean