Lesson

BA501 : The Class : Stats : Measures : Introduction
Descriptive Measures: Introduction
Introduction

§ Descriptive Measures §

Lesson 1: Introduction

Descriptive measures are designed to give a single number or range of numbers to help understand a given set of data. The measures are categorized as Measures of Central Tendency, Measures of Dispersion, Measures of Position, and Measures of Shape. You will study a selected number of measures following into the first three of these categories. In addition, you will consider the Empirical Rule. In most cases, you will be able to use Excel to aid you in this process. Click the equation button to the left which will reveal these measures and others not covered at this web site.

§ Sample Mean § Population Mean §

§ Mode § Median § Sample Variance §

§ Population Variance § Z Score § Empirical Rules §

§ Examples §

Measures of Central Tendency

Sample Mean

The sample mean is also known as the average. The mean can be viewed as a point of balance, but not necessarily located at the center of the data set. The sample mean is calculated from a simple random sample taken from a population. It is a statistic based on the simple random sample, and used as estimate a parameter (the average in the population). (click me)

When is an average located at the center of the data? When the data are: symmetric non-symmetric (click one)

In a bell-shaped distribution (normal distribution), the mean is located in the _________ of the distribution.

Population Mean

The population mean is the average of all the data contained in a population. A census is taken when the entire population is considered. If the population parameter is known, then no inference is necessary by calculating a statistic from a simple random sample. All the population data is seldom known or is unknowable. Even if the entire population is known, it may be too costly to survey each member of the population. In some cases, the sampling process actually destroys the items being sampled. (click me)

If the population mean is know, then a sample mean is not necessary to draw inferences about the population. True False (click one)

A population is the set of all ______ of interest.

Mode

The mode is the most frequent (highest count) data value in the data set. When all values have the same frequency, then the data has no mode. Some data sets are bimodal when two data values have the same count. (click me)

Where is the mode for a normal (bell-shaped) distribution of numbers? Middle No Mode (click one)

The mode is a measure of __________ tendency.

Median

The median is the middle data value in a data set when the data are placed in an ordered array. An ordered array is obtained by placing the data in order from lowest to highest. If the number of data values is odd, then the median is the middle data value which has an equal number of data values on each side. When the number of data values is even, then the median is the average of the two middle values in the ordered array. (click me)

The median can be made up of more than two numbers. True False (click one)

When there is an _________ in the data, the median may be a better measure of central tendency than the mean.

Measures of Dispersion

Sample Variance

The sample variance is a measure of the dispersion of sample data values around their mean. The sample variance is in squared units, so it is usually adjusted to the sample standard deviation by taking its square root. The sample variance is the average (dividing by n -1) of the sum of the squared deviations of all the x data values from their mean. The sample variance, s², is a statistic used to estimate the population variance, s². (click me)

What does a sample variance measure? The spread of the data. The center of the data. (click one)

The variance is in __________ units.

Population Variance

The population variance is a measure of the dispersion of population data values around their mean. The population variance is in squared units, so it is usually adjusted to the population standard deviation by taking its square root. The population variance is the average (dividing by N) of the sum of the squared deviations of all the x data values from their mean, m. The population variance, s², is a parameter. If the population parameter is known, then no inference is necessary by calculating a statistic from a simple random sample. (click me)

The population variance is calculated by dividing the squared deviations by the sample size. True False (click one)

The population variance is a(n) ____________ based on a census.

Measures of Position

Z Score

A Z score measures the relative position of a data value from its mean. It measures the number of standard deviations these two values are apart. Z scores are dimensionless unit measure, no matter what units of measure are used in the data set. It is the individual deviation (x - xbar) divided by the standard deviation (average deviation). (click me)

A Z score has the same units of measure as the slope for the same data. True False (click one)

When a data value, x, is equal to the mean, xbar, what is the Z score?

Empirical Rules

The empirical rules are used to measure the percentage of data values between two points a normal (bell-shaped) distribution. These end points are fixed at one standard deviation (68% rule) , two standard deviations (95% rule) and three standard deviations (99.7% rule) on each side of xbar. (click me)

The pecentages represented in the three Empirical Rules are the only percentages possible between two points in a normal (bell-shaped) distribution. True False (click one)

The 99.7% Empirical Rule means that _______ percent of the data is not captured in each end of the normal distribution.

Examples: See link below.

You should now:

Go to Lesson 2: Examples

Go back to Descriptive Measures: Assignments and Activities

Please reference "BA501 (your last name) Assignment name and number" in the subject line of either below.

E-mail Dr. James V. Pinto at BA501@mail.cba.nau.edu
or call (928) 523-7356. Use WebMail for attachments.