MATH IISTATISTICS

REVIEWStatistics topics from both Math 1 and Math 2, both featured on the

GHSGT

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 The population mean as 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 =

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 there are two or more numbers that occur the SAME NUMBER of times, then there is “NO 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.

PROCEDURES…1. Put distribution in ascending order.2. Lower extreme – the lowest value3. Upper extreme – the highest value4. Find the median5. From the lower extreme to the median is the lower quartile. Find the median of the lower quartile. This is called Q1.6. From the median to the upper extreme is called the upper quartile. Find the median of the upper quartile. This is called the Q3.

PROCEDURES FOR FINDING THE 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.

PROCEDURES 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…

VARIANCE(FOR A POPULATION)

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 size

FORMULA… The formula for variance computed in an entire population is

where σ2 represents the variance, μ is the mean, and N is the number of scores.

VARIANCE(FOR A SAMPLE)

When computed in a sample in order to estimate the variance in the population, the formula is where s2 is the estimate of variance, X is the sample mean, and N is the number of scores in the sample.

FORMULA…

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.

FORMULA…

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 µ is the mean and σ is the standard deviation.