Measures of Central Tendency

Numbers or words that attempt to describe, most generally, the middle or typical value for a distribution.

Mode

The value of the most frequent score.

Bimodal

Describes any distribution with two obvious peaks.

Multimodal

Describes any distribution with more than two peaks.

Determine the mode for the following retirement ages:

60, 63, 45, 63, 65, 70, 55, 63, 60, 65, 63

Mode = 63

The owner of a new car conducts six gas mileage tests and obtains the following results, expressed in miles per gallon. Find the mode for these data.

26.3, 28.7, 27.4, 26.6, 27.4, 26.9

Mode = 27.4

Median

The middle value when observations are ordered from least to most.

What are the 6 steps for finding the median?

- Order scores from least to most.
- Find the middle position by adding one to the total number of scores and dividing it by 2.
- If the middle position is a whole number, use this number to count into the set of ordered scores.
- The value of the median equals the value of the score located at the middle position.
- If the middle position is not a whole number, use the two nearest whole numbers to count into the set of ordered scores.
- The value of the median equals the value midway between those of the two middlemost scores; to find the midway value, add the two given values and divide by 2.

Find the median for the following retirement ages:

60, 63, 45, 63, 65, 70, 55, 63, 60, 65, 63

Median = 63

Find the median for the following gas mileage tests:

26.3, 28.7, 27.4, 26.6, 27.4, 26.9

Median = 27.15

How do you find the mean?

The mean is found by adding all scores and then dividing by the number of scores.

Population

A complete set of scores.

Sample

A subset of scores.

Sample Mean ( ̅X)

The balance point for a sample, found by dividing the sum for the values of all scores in the sample by the number of scores in the sample.

̅X = (∑X) / n

Sample Size (n)

The total number of scores in the sample.

Population Mean (µ)

The balance point for a population, found by dividing the sum for all scores in the population by the number of scores in the population.

µ = (∑X) / N

Population Size (N)

The total number of scores in the population.

Find the mean for the following retirement ages:

60, 63, 45, 63, 65, 70, 55, 63, 60, 65, 63

Mean = 672 / 11 = 61.09

Find the mean for the following gas mileage tests:

26.3, 28.7, 27.4, 26.6, 27.4, 26.9

Mean = 163.3 / 6 = 27. 22

Indicate whether the following skewed distribution are positively skewed or negatively skewed:

a distribution of test scores on an easy test, with most students scoring high and a few students scoring low

negatively skewed because the median exceeds the mean

Indicate whether the following skewed distribution are positively skewed or negatively skewed:

a distribution of ages of college students, with most students in their late teens or early twenties and a few students in their fifties or sixties

positively skewed because the mean exceeds the median

Indicate whether the following skewed distribution are positively skewed or negatively skewed:

a distribution of loose change carried by classmates, with most carrying less than $1 and with some carrying $3 or $4 worth of loose change.

positively skewed because the mean exceeds the median

Indicate whether the following skewed distribution are positively skewed or negatively skewed:

a distribution of the sizes of crowds in attendance at a popular movie theater, with most audiences at or near capacity

negatively skewed because the median exceeds the mean

College students were surveyed about where they would most like to spend their spring break: Daytona Beach (DB); Cancun Mexico (C), South Padre Island (SP), Lake Havasu (LH), or other (O). The results were as follows:

DB DB C LH DB

C SP LH DB O

O SP C DB LH

DB C DB O DB

Find the mode and the median.

Mode = DB (Daytona Beach)

Impossible to fin the median when qualitative data are unordered, with only nominal measurement.