Slide 3-2

Chapter 3Descriptive Measures

Section 2 Measures of Variations

Slide 3-3

Measures of Variation

There are three common measures for the spread or variability of a data set they are range, variance, and standard deviation.

To describe the difference quantitatively, we use a descriptive measure that indicates the amount of variation, or spread, in the data set. These are referred to as measures of variation or measures of spread.

The range rule of thumb: A rough estimate of the standard deviation is

range

4s

Slide 3-44

Variance:

is the average of the squares of the distance each value is from the mean.

The symbol for the population variance is σ2. Greek lower case letter sigma.

Symbol for sample variance is s2.

N

X

2

2 Population Variance =

2

2

1

X Xs

N

Sample Variance =

X = individual values

μ = population mean

N = population size

= sample mean

n = sample size

X

Slide 3-5

Figure 3.3

The “data sets” have the same Mean, Median, and Mode yet clearly differ!

Measures of Variation or Measures of Spread

Slide 3-6

Range of a Data SetRange of a Data Set

The range of a data set is given by the formula

Range = Max – Min,

where Max and Min denote the maximum and minimum observations, respectively.

Range:

distance between the highest value and the lowest value.

The symbol R is used for the range.

R = highest value – lowest value

Slide 3-7

Figure 3.4

Measures of Variation or Measures of Spread:The Range

Team I has range 6 inches, Team II has range 17 inches.

Slide 3-8

Standard Deviation:

Measures variation by indicating how far, on average, the

observations are from the mean.

is the square root of the variance.

Symbol for the population standard deviation is σ (sigma). Symbol

for the sample standard deviation is s.

Variation and the Standard Deviation: The more variation

that there is in a data set the larger is its standard deviation

Rounding rule: The final answer should be rounded to one more

decimal place than the original data.

2

2 X

N

Population Standard Deviation =

Sample Standard Deviation = 2

2

1

X Xs s

n

Slide 3-9

Sample Standard Deviation

Deviations from the Mean is how far each observation is from the mean and is the first step in computing a sample standard deviation.

Sum of Squared Deviation is the sum of the squared deviations from the mean ∑(x1 - )2 and gives a measure of the total deviations from the mean for all the observation.

x

Slide 3-10

Sample Standard Deviation – standard deviation of a sample. Take the square root of the sample variance.

1

)( 22

2

n

xxss

Why is the denominator (n – 1) rather than n?

Division by (n – 1) increases the value of the sample variance so that it will more closely reflect the population variance. Giving us an unbiased estimate for the population variance.

Shortcut or computational formulas for data obtained from samples:

1

/)( 222

n

nXXs

Variance Standard Deviation

1

/)( 22

n

nXs X

Sample variance formula =

Slide 3-11

Variance and Standard Deviation

Variances and standard deviations can be used to determine

the spread of the data. If the variance or standard deviation is

large, the data are more dispersed. The information is useful

in comparing two (or more) data sets to determine which is

more (most) variable.

The measures of variance and standard deviation are used to

determine the consistency of a variable.

The variance and standard deviation are used to determine

the number of data values that fall within a specified interval

in a distribution.

The variance and standard deviation are used quite often in

inferential statistics.

Slide 3-12

Computing Formula for a Sample Standard Deviation

Rounding rule: do not perform any rounding

until the computation is complete, otherwise,

substantial round off error can result.

Slide 3-13

Table 3.10

Table 3.11

Standard Deviation: the more variation, the larger the standard deviation. Data set II has greater variation.

Slide 3-14

Figure 3.6

Data set II has greater variation and the visual clearly shows that it is more spread out.

Data Set I

Figure 3.7

Data Set II

Slide 3-15

Almost all the observations in any data set lie within three standard deviations to either side of the mean.

Three-Standard-Deviation Rule

Slide 3-16

Example:Find the variance and standard deviation.

Exam Scores: For 108 randomly selected college students, the exam score frequency distribution was obtained.

Class limits

f Xm Midpoint

f • Xm f • (Xm)2

90-98 6 94 564 53,016

99-107 22 103 2266 233,392

108-116 43 112 4816 539,392

117-125 28 121 3388 409,948

126-134 9 130 1170 152,100

108 12,204 1,387,854

(90+98)/2 94

6 * 94 564

6*(94)2 53,016

Slide 3-17

82.3or 26.821108

108204,12

854,387,12

2

s

9.1or 07.926.82 sStandard Deviation =

Variance =

*** NOTE: 9.072 = 82.26 ***

Example: