MEASURES OF CENTRAL TENDENCY

MEAN The mean is typically what is meant by

the word “average.” The mean is perhaps the most common measure of central tendency.

The sample mean is written as (x bar). The population mean is written with the

Greek letter mu (μ). Despite its popularity, the mean may not

be an appropriate measure of central tendency for skewed distributions, or in situations with outliers.

MEAN CONTINUED… PROCEDURES…

1. Add up the values (X) of the distribution.2. Divide the sum by the total number (N) in

the distribution.

FORMULA…

Mean… X or µ =

What is this?

∑

MEDIAN The median is a popular measure of

central tendency. To find the median of a number of

values, first order them, then find the number in the middle. (Note that if there is an even number of values, one takes the average of the middle two.)

The median is often more appropriate than the mean in skewed distributions, or in situations with outliers.

MODE The mode is the most common value in

a distribution. Note that the mode may be very

different from the mean and the median.

If every number is listed only once (or ALL are listed the same number of times, then there is “NO MODE.”

If several numbers are listed more than once and listed the same number of times, then all those numbers are the mode.

BOX AND WHISKER PLOTS

BOX AND WHISKER PLOTS A box-and-whisker plot can be useful for handling

many data values. They allow people to explore data and to draw informal conclusions when two or more variables are present.

It shows only certain statistics rather than all the data.

Five-number summary is another name for the visual representations of the box-and-whisker plot. The five-number summary consists of the median, the quartiles, and the smallest and greatest values in the distribution.

Immediate visuals of a box-and-whisker plot are the center, the spread, and the overall range of distribution.

PROCEDURESFOR FINDING FIVE NUMBERS

(Put distribution in ascending order)

1. Lower extreme – the lowest value2. Upper extreme – the highest value3. Find the median4. The set of all the numbers that are lower than

the median are called the lower quartile. Find the median of the lower quartile.

This is called Q1.

5. The set of numbers that are above the median is called the upper quartile. Find the median of the upper quartile.

This is called the called the Q3.

STEPS FOR GRAPHING BOX PLOT

1. First you will need to draw an ordinary number line that extends far enough in both directions to include all the numbers in your data.

2. Locate the main median using a vertical line just above your number line.

3. Locate the lower median (Q1) and the upper median (Q3) with similar vertical lines.

4. Draw a box using the lower and upper median lines as endpoints.

5. The whiskers extend out to the data's smallest number and largest number.

*The interquartile

range is…Q3 – Q1

Q1

Q1 Q3

MEASURES OF VARIABILITY

RANGE The difference between the maximum

and minimum values of a distribution. The range is the simplest measure of

variability.

PROCEDURE…1. Take the largest value and subtract the smallest value.

FORMULA…High – Low = Range

INTERQUARTILE RANGE (IQR)

The Interquartile Range (IQR) is the(75th percentile – 25th percentile).

PROCEDURE…1. Find Q3 and Q1 of a data set.

2. Subtract Q1 from Q3.

FORMULA…Q3 – Q1 = IQR

MEAN ABSOLUTE DEVIATION(MAD)

It is used as a measure of variability where the number of values or quantities is small, otherwise standard deviation is used.

PROCEDURES…1. Find the mean (X) of the data. 2. Subtract the mean from each data value to get the

deviation from the mean. 3. Take the |absolute value|of each deviation from the

mean. 4. Total (∑) the absolute values of the deviations from

the mean.5. Divide the total by the sample size (N).

FORMULA… MAD =

VARIANCE The variance is a widely used measure of

variability. It is defined as the mean squared deviation of scores from the mean.

PROCEDURE…1. Find the mean of the data. 2. Subtract the mean from each value to find the

deviation from the mean. 3. Square the deviation from the mean. 4. Total the squares of the deviation from the

mean. 5. Divide by the population (N) or if a sample,(N–

1).

VARIANCE(FOR A SAMPLE)

The formula for variance computed in a sample distribution is

FORMULA… s2 =

Where X is the a value in the distributionX (x-bar) is the mean of the sampleN is the number in the sample s2 is the variance of the sample

VARIANCE(FOR A POPULATION)

The formula for variance computed in an entire population is

FORMULA…

Where X is the a value in the distributionµ (mu) is the mean of the populationN is the population σ2 is the variance of the population

STANDARD DEVIATION The standard deviation is a widely used

measure of variability. An important attribute of the standard

deviation as a measure of variability is that if the mean and standard deviation of a normal distribution are known, it is possible to compute the percentile rank associated with any given score.

PROCEDURES…1. Find the variance of the distribution.2. Take the square root of the variance.

STANDARD DEVIATION (FOR SAMPLE)

FORMULA… s =

Where X is the a value in the distributionX (x-bar) is the mean of the sampleN is the number in the sample s is the standard deviation of the sample(It’s the square root of the variance!)

STANDARD DEVIATION (FOR POPULATION)

FORMULA…

Where X is the a value in the distributionµ (mu) is the mean of the populationN is the population σ is the standard deviation of the population(It’s the square root of the variance!)

NORMAL DISTRIBUTION

NORMAL DISTRIBUTION One of the most common continuous

distributions, a normal distribution is sometimes referred to as a "bell-shaped distribution."

A graph of a normal distribution is shown below.

EMPIRICAL RULE The rule states…

Approximately 68% of the values will lie within one standard deviation of the mean.

Approximately 95% of the values will lie within two standard deviations of the mean.

Approximately 99.7% of the values will lie within three standard deviations of the mean.

The empirical rule is sometimes called the "68-95-99.7 Rule".

Where σ is the standard deviation from the mean.

DISTRIBUTION The distribution of empirical data is

called a frequency distribution and consists of a count of the number of occurrences of each value.

SAMPLE A sample is a subset of a population, often taken

for the purpose of statistical inference. Generally, one uses a random sample.

See also: Bias - A sampling method is biased if each element

does not have an equal chance of being selected.

Stratified random sample - In stratified random sampling, the population is divided into a number of subgroups (or strata). Random samples are then taken from each subgroup with sample sizes proportional to the size of the subgroup in the population.

POPULATION A population is the complete set of

observations a researcher is interested in. Contrast this with a sample which is a subset of a population.

A population can be defined in a manner convenient for a researcher.

For example, one could define a population as all girls in fourth grade in Houston, Texas. Or, a different population is the set of all girls in fourth grade in the United States.

Inferential statistics are computed from sample data in order to make inferences about the population.

OUTLIERS Outliers are atypical, infrequent

observations; values that have an extreme deviation from the center of the distribution.

There is no universally-agreed on criterion for defining an outlier, and outliers should only be discarded with extreme caution.

However, one should always assess the effects of outliers on the statistical conclusions.

SIGMA NOTATION Many statistical formulas involve

summing numbers. Fortunately there is a convenient notation for expressing summation.

When no values are shown, it means to sum all the values of X.