Module 2 Notes

Page 1

Measures of Central Location:

Median - If the data set contains an odd number of values, the median is one of the values. If there is an even number of values in the data set, the median is not necessarily one of the values. For example, if there are 10 values in a data set, the median is the 5.5th value. That is, it falls halfway between the fifth and sixth values in the ordered data set.

Greek letters identify population parameters, and lower-case Roman letters identify sample statistics.

Mean:

referred to as the balancing point

The main disadvantage of using the arithmetic mean as a measure of

central location is the degree to which one very large or very small

value in the data set can influence the value of the arithmetic mean

Mode:

is the most frequently occurring observation in a data set. some data sets have no mode, some have more than one.

geometric mean (GM) is a useful measure of central

location, particularly in accounting and finance. The geometric mean

measures the average rate of change (growth or decline) in a quantity.

It is calculated as the nth root of the product of n quantities, each

representing a growth factor. A growth factor is calculated by adding 1

to the rate of change for a period of time. For example, if a bond’s

value grew at 5% for a year, the growth factor would be 1.05. If sales

fell by 10% in one year, the growth factor would be (1.0 – 0.10), or

0.90. The formula for calculating the geometric mean is

Rg = [(1 + R1)(1 + R2) … (1 + Rn)]1/n – 1

where n = the number of growth factors

Measures of variability:

range:

simple measure of the total spread of the data. does not take into consideration extreme values

=high value - low value

variance:

measures how far the indiviual value in a data set are spread out from the mean

difference between population and sample formula: To arrive at the population variance, you divide by N (the number of

items in the population). To arrive at the sample variance, you divide

by (n – 1) (the number of items in the sample minus one).

There are two important characteristics of the variance as a measure of dispersion:

It is independent of the mean of the population.

It is measured in terms of the squares of deviations.

standard deviation:

is the square root of the variance - better unit of measure than the variance

coefficient

of variation:

measures variability as a percentage

=standard deviation\mean

Approximating the variance and standard deviation for grouped data need to find the midpoints, fm and fm2 and use a different formula:

Measures of relative standing

A box plot is a pictorial representation of the

summary values in a data set. The values used to construct the box plot

(or box and whisker plot) are the smallest and largest values in the

data set, the first and third quartiles, and the median (second

quartile). In the box plot diagram in Figure 4.1 on page 121, the

whiskers emanate from the box indicating 25% of the data on each side

of the box. The box represents the inter-quartile range as it shows the

distance between the first and third quartiles.

Measures of association Covariance:

is a measure of how two variables change together

he population and sample covariance formulas on page 127 are similar to

the formulas for the population and sample variances. Instead of

squaring the difference between each x value and the mean of the x

values (the first variable), the difference is multiplied by the

difference between each corresponding y value and the mean of the y

values (the second variable)

variance is never negative since the difference between each x value and

the mean is squared. However, the covariance can be negative

The covariance is negative if the value of x is increasing while the

value of y is decreasing. That is, there will be a negative covariance

if there is an inverse relationship between the two variables. A

negative relationship is seen on the scatter plot if the data tend to

follow a straight line from the upper left corner of the graph to the

bottom right corner of the graph. If there is very little linear

relationship between the variables, the covariance will be close to

zero.

coefficient of correlation to measure the degree of association between two variables. The coefficient of correlation (ρ or r) = covariance ÷ standard deviations of x and y (see page 128).

It has a value between –1 and +1. A value close to –1 indicates a

strong negative, or inverse, linear relationship between the variables,

while a value close to +1 indicates a strong positive linear

relationship. A value close to zero indicates the lack of a linear

relationship. However, just because there is a linear relationship

between two variables does not necessarily mean that one causes the

other. This important concept is explained on page 141.

The least squares method is a means of plotting a straight

line through the scatter points that best represents the linear relationship

between the variables. While the coefficient of correlation explains the

strength of the relationship, the least squares line tells you what the

linear relationship is between the variables.

The equation for a straight line (see page 132) is

where b0 = the y-intercept and b1 = the slope of the line.

The formulas for determining the slope and the y-intercept are on

page 132. Note that the numerator in the formula for the slope of the

line is

the covariance.