MEASURES OF VARIANCE

2.4

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Why we need to learn something so we never sound like this.

RANGE The simplest measure of variance is the

range. The range of a data set is the difference

between the maximum and minimum data entries in the set.Range = (maximum data entry)-(Minimum data entry)

Midrange = (maximum data entry)-(Minimum data entry)/2

FINDING THE RANGE OF A DATA SET Two corporations each hired 10 graduates. The

starting salaries for each are shown. Find the range of the starting salaries for Corporation A and B.

Starting Salaries for Corporation A (1000’s of Dollars)

Salary 41 38 39 45 47 41 44 41 37 42

Starting Salaries for Corporation B (1000’s of Dollars)

Salary 40 23 41 50 49 32 41 29 52 58

Ordering the data helps to find the least and greatest salaries.37 38 39 41 41 41 42 44 45 47Range = (Maximum salary) – (Minimum salary)= 47 – 37=10Find Corporation B

DEVIATION, VARIANCE, AND STANDARD DEVIATION As a measure of variation, the range has

the advantage of being easy to compute. Its disadvantage, however, is that it uses only two entries from the data set. Two measures of variation that use all the entries in a data set are the variance and the standard deviation. However, before you learn about these measures of variation, you need to know what is meant by the deviation of an entry in a data set.

DEVIATION The deviation of an entry x in a population

data set is the difference between the entry and the mean μ of the data set.

Deviation of x = x – μ Find the deviation in Corporation A Starting Salaries for Corporation A (1000’s of Dollars)

Salary 41 38 39 45 47 41 44 41 37 42

Starting Salaries for Corporation B (1000’s of Dollars)

Salary 40 23 41 50 49 32 41 29 52 58

The mean starting salary is μ = 415/10 = 41.5. To find out how much the salary deviates from the mean, subtract 41.5 from the salary. For instance, the deviation of 41 (or $41,000) is41 – 41.5 = -0.5 or (-$500)Find the deviation of salary $49,000

FINDING DEVIATIONS OF A DATA SET Notice that the sum of

the deviations is zero. Because this is true for any set, it doesn’t make sense to find the average of the deviations. To overcome this problem, you can square each deviation. In a population data set, the mean of the square of the deviations is called the population variance.

Salary(1000s of dollars)

x

Deviation(1000s of dollars)x - μ

41 -0.538 -3.539 -2.545 3.547 5.541 -0.544 2.541 -0.537 -4.542 -0.5

Ʃx = 415 Ʃ(x-μ) = 0

VARIANCE Variance is often used as a measure of risk in finance. It

is a statistical concept whose purpose is to measure the dispersion (the spread of the #’s) of observation around the mean or the average value.

Risk is the possibility that actual returns might differ, or vary, from expected returns. In fact, actual returns will likely differ from expected returns. It is important for decision-makers to estimate the magnitude and likelihood of the difference between actual and estimated returns. After all, there is a big difference if your predictions result in an error of only $100 versus an error of $1 million.

By using the concepts of variance and standard deviation, investors can judge not only how wrong their estimates might be, but also estimate the likelihood, or probability, of favorable or unfavorable outcomes. With the tools of expected return and standard deviation, financial decision-makers are better able to evaluate alternative investments based on risk-return tradeoffs, and their own risk preferences.

STANDARD DEVIATION AND VARIANCE The preceding computational procedure

reveals several things. First, the square root of the variance gives the standard deviation; and vice versa, squaring the standard deviation gives the variance.

Second, the variance is actually the average of the square of the distance that each value is from the mean. Therefore, if the values are near the mean, the variance will be small. In contrast, if the values are far from the mean, the variance will be large.

Remember the sum of the distances will always be zero!!!!

POPULATION VARIANCE The variance is the average of the

squares of the distance each value is from the mean.

Population variance =

σ2 is the lowercase Greek letter sigma X=individual value

μ= population value

N= population size

The population standard deviation of a population data set of N entries is the square root of the population variance.Population standard deviation =http://www.youtube.com/watch?v=jYd38gALa04

FINDING THE POPULATION VARIANCE AND STANDARD DEVIATION

FINDING THE POPULATION STANDARD DEVIATION

Find the population variance and standard deviation of the starting salaries for Corporation A given in Example 1. μ = 41.5

The table at the left summarizes the steps used to find SSx = 88.5

N= 10 Population Variance = 8.9 Population Standard

deviation = 2.98 or $2,999.

Salary(1000s of dollars)

x

Deviation

(1000s of

dollars)x - μ

Squares(x – μ)2

41 -0.5 0.2538 -3.5 12.2539 -2.5 6.2545 3.5 12.547 5.5 30.2541 -0.5 0.2544 2.5 6.2541 -0.5 0.2537 -4.5 20.2542 -0.5 0.25

Ʃx = 415 Ʃ(x-μ) = 0

SSx=88.5

INTERPRETING STANDARD DEVIATION When interpreting the standard deviation,

remember that it is a measure of the typical amount an entry deviates from the mean. The more the entries are spread out, the greater the standard deviation.

SAMPLE VARIANCE AND SAMPLE STANDARD DEVIATION The sample variance formula and

sample standard deviation formula of a sample data set of n entries are listed below.

Sample variance =

Sample Standard Deviation =

FINDING SAMPLE VARIANCE AND STANDARD DEVIATION

SHORTCUT FOR SAMPLE DATA

PRACTICE Find the sample variance and standard deviation for the

amount of European auto sales for a sample of 6 years shown, The data are in millions of dollars.

11.2, 11.9, 12.0, 12.8, 13.4, 14.3

SUMMARY AND FORMULAS