# DESCRIPTIVE MEASURES

Document Sample

Perform Descriptive
Measures Statistical
Procedures

FINANCE SCHOOL       805A-44A-1033-VG-1
Terminal Learning Objective

Action: Perform descriptive measures statistical
procedures by computing the measures of central
tendency and dispersion; constructing a frequency
distribution (absolute, relative, and cumulative);
plotting a histogram, frequency polygon, and
cumulative frequency distribution; and identify the
skewness of the direction.

Conditions: Given the Formulas provided in the Summary
Sheet and a data set (population or sample).

Standard: IAW the Descriptive Measures Programmed Text
and Accepted Statistical Techniques.

FINANCE SCHOOL                    805A-44A-1033-VG-2
What is (or are) Statistics?

“There are Three Types of Lies: LIES, DAMNED LIES,
and STATISTICS.” -- DISRAELI. Other quotes are:

There was a Man who Drowned in a River only 2 Feet Deep
--- On Average, That Is.

People use Statistics as a Drunk uses a Lamppost
--- More for Support than for Illumination.

Statistics are like a Razor
--- Vital, Variable, And Vicious!

Figures Don’t Lie, But Liars Figure!

FINANCE SCHOOL              805A-44A-1033-VG-3
Descriptive Terms

• Population (N): is the whole thing.

• Sample (n): is a part of the whole.

• Parameter.

• Statistic.

FINANCE SCHOOL          805A-44A-1033-VG-4
Sampling Procedures

Select a Sample

Population                       Sample
Parameter                        Statistic
Make an Inference

FINANCE SCHOOL         805A-44A-1033-VG-5
Measures of
Central Tendency

Mean.

Median.

Mode.

FINANCE SCHOOL    805A-44A-1033-VG-6
MEAN

•   Population Mean ():  = X       Where:

   X = observations
 = arithmetic
population mean
N = # observations

Where:
•   Sample Mean (X): X = X
X = observations
n
X = arithmetic
population mean
n = # observations

FINANCE SCHOOL              805A-44A-1033-VG-7
Population

 = X = 7 + 9 + 10 + 10 + 11 + 13 = 60 = 10
N            6                6

FINANCE SCHOOL                 805A-44A-1033-VG-8
Extreme Values

 = 7 + 9 + 10 + 10 + 11 + 13 + 31 = 91 = 13
        7                     7

FINANCE SCHOOL                 805A-44A-1033-VG-9
Characteristics
of the Mean

• Familiar to most people.

• Always exists.

• Always unique.

• Lends itself to statistical manipulation.

• Accounts for each individual item.

• Reliable estimator.

FINANCE SCHOOL             805A-44A-1033-VG-10
MEDIAN

The middle item (or mean of the two middle items)
found within a set of data, when the data is
arranged in order of magnitude.

FINANCE SCHOOL             805A-44A-1033-VG-11
Characteristics
of the Median

• Always exists.

• Always unique.

• Ordered array required.

• Less reliable than the mean as an estimator.

FINANCE SCHOOL            805A-44A-1033-VG-12
MODE

The most frequently occurring observation or
observations in the data.

X            X             X
7            7             7
9            9             9
10           10             9
]
13           10
]          10
16           13            13
17           15            13
]
No Mode      Unimodal      Bimodal

FINANCE SCHOOL             805A-44A-1033-VG-13
Characteristics
of the Mode

• May not always exist.

• Requires no calculation.

• Are not unique.

FINANCE SCHOOL    805A-44A-1033-VG-14
Arithmetic Mean

Population Mean = 

Sample Mean = X

FINANCE SCHOOL   805A-44A-1033-VG-15
Measures of Dispersion

Range.

Variance.

Standard Deviation.

FINANCE SCHOOL       805A-44A-1033-VG-16
RANGE

The difference between the two extreme values of a
set of a data.

RANGE = LARGEST VALUE - SMALLEST VALUE

Rg = Xmax - Xmin

FINANCE SCHOOL              805A-44A-1033-VG-17
Average Deviation

Each data point of the data set differs from the mean by
a measurable amount. We can measure this in a
general sense by the formula:
di = Xi - 
where di is the difference between any specific data
point, Xi , and the mean.

FINANCE SCHOOL                 805A-44A-1033-VG-18
Original Data Set

Let’s consider our original data set:

7, 9, 10, 10, 11, 13

the mean is:

 = 7 + 9 + 10 + 10 + 11 + 13 = 60 = 10
6              6

FINANCE SCHOOL         805A-44A-1033-VG-19
Data Point

Now we find the deviation of each data point from the mean:
X        X-  = d
7        7 - 10 = -3
9        9 - 10 = -1
10       10 - 10 = 0
10       10 - 10 = 0
11       11 - 10 = +1
13       13 - 10 = +3
 (X - ) = 0
N

For this and any data set, the average of the deviations
FINANCE SCHOOL                 805A-44A-1033-VG-20
VARIANCE

The mean of the squares of the variations from the
mean of a frequency distribution.
2
x)
2
x - N
2

Population Variance:      
N
2
x)
2
x - n
2

Sample Variance:          S
n-1

FINANCE SCHOOL                805A-44A-1033-VG-21
Standard Deviation

The square root of the mean of the standard deviations.

2
Population Std Deviation:                 x)
x - N
2

     2

N

Sample Std Deviation:                            2
x)
2
x - n
2

S   S
n-1

FINANCE SCHOOL              805A-44A-1033-VG-22
Squared Deviations

X          X-  = d       (X - )2
7          7 - 10 = -3        9
9          9 - 10 = -1        1
10         10 - 10 = 0         0
10         10 - 10 = 0         0
11         11 - 10 = +1        1
13         13 - 10 = +3        9
 X = 60    (X - ) = 0        20
N

FINANCE SCHOOL              805A-44A-1033-VG-23
Mean Squared Deviation

The average or mean squared deviation is found
by dividing the sum of the squared deviations by
the number of observations:

(X - )2 = 20 = 3 1/3
N        6
The mean squared deviation is also called the
variance, and is denoted by the lower case Greek
letter (sigma) squared: 2

FINANCE SCHOOL              805A-44A-1033-VG-24
Alternative Formulation

An alternative formulation for both the variance and
standard deviation simplifies the work involved. These
are:
(X)2
Variance: 2 = (X - )2 = X2 - N
N            N

(X)2
Standard Deviation = =       X2 - N
N

FINANCE SCHOOL                805A-44A-1033-VG-25
Worksheet A

Worksheet A for 2 =  (X - )2
N

X          X-  = d          (X - )2
7          7 - 10 = -3           9
9          9 - 10 = -1           1
10         10 - 10 = 0            0
10         10 - 10 = 0            0
11         11 - 10 = +1           1
13         13 - 10 = +3           9
 X = 60      (X - ) = 0          20
N

 = 60 = 10        2 = 20 = 3.3333    = 3.3333 = 1.8257
6                   6

FINANCE SCHOOL               805A-44A-1033-VG-26
Worksheet B
(X)2
Worksheet B for 2 = X2 - N
N
X                   X2
7                   49
9                   81
10                  100
10                  100
11                  121
13                  169
X    =       60     X2     =    620

(60)2
2 = 620 - 6 = 620 - 600 = 20 = 3.3333
6         6         6

= 3.3333 = 1.8257

FINANCE SCHOOL                 805A-44A-1033-VG-27
Coefficient of Variation

C = 100(/)

- standard deviation is a measure of absolute variability.

- coefficient of variation is a measure of relative
variability.

Group A                Group B
 = \$6.61                   \$5.82
=   .57                    .51
C = 8.62%                    8.76%

FINANCE SCHOOL                     805A-44A-1033-VG-28
Federalist Papers

0.4
Hamilton
0.35
0.3
Proportion of Papers

Disputed
0.25

0.2

0.15

0.1

0.05

0
0   1   3   5     7    9   11    13   15   17   19

Rate of Usage

FINANCE SCHOOL                       805A-44A-1033-VG-29
Pure Chemical Company

Output of 10 curing VATs 19 Jan **.

VAT         Gallons Produced
A                65
B                67
C                66
D                68
E                66
F                67
G                66
H                65
I               64
J                68
FINANCE SCHOOL           805A-44A-1033-VG-30
Frequency Distribution
(Pure Chemical Company)

VAT output 19 Jan **.
Frequency
Output Gallons    Absolute   Relative
64              1      1/10 = 0.1

65             2       2.10 = 0.2

66             3       3.10 = 0.3

67             2       2/10 = 0.2

68              2       2/10 = 0.2
TOTALS              10              1.0
FINANCE SCHOOL           805A-44A-1033-VG-31
Histogram
(Pure Chemical Company)

Curing VAT output 19 Jan **.
3                                                         0.3

Relative Frequency
Absolute Frequency

2                                                         0.2

1                                                         0.1

0                                                         0
0    64    65         66           67   68   0
Output (Gallons)

FINANCE SCHOOL                         805A-44A-1033-VG-32
Frequency Polygon
(Pure Chemical Company)

Curing VAT output 19 Jan **.
3                                                        0.3

Relative Frequency
Absolute Frequency

2                                                        0.2

0.1
1

0                                                        0

0     64     65         66           67   68    0
Output (Gallons)

FINANCE SCHOOL                      805A-44A-1033-VG-33
Population Test
N = 34

ROSTER            ROSTER
1        98       18       99
2        84       19       78
3        86       20       99
4       100       21       86
5        94       22       95
6        97       23      100
7        81       24       93
8        86       25       97
9        99       26       97

FINANCE SCHOOL      805A-44A-1033-VG-34
Population Test
N = 34 (Cont)

ROSTER            ROSTER
10       87       27       79
11       86       28       91
12       90       29       88
13       89       30       93
14       96       31       94
15       94       32       84
16       98       33       89
17       90       34       91

FINANCE SCHOOL      805A-44A-1033-VG-35
Frequency Distribution
for Test Scores
FREQUENCY          .
MIDPOINT   CLASS INTERVAL   TALLY   ABSOLUTE RELATIVE CUMULATIVE
80         78-81                    3       .088       .088
84         82-85                    2       .059       .147
88         86-89                    8       .235       .382
92         90-93                    6       .176       .558
96         94-97                    8       .235       .793
100         98-100                   7       .206       .999
34      .999*
* NOT EXACT DUE TO ROUNDING.

FINANCE SCHOOL                    805A-44A-1033-VG-36
Test Score Histogram

8
Absolute Frequency

6

4

2

0
78   82    86    90     94   98   102

FINANCE SCHOOL                805A-44A-1033-VG-37
Test Score Frequency Polygon

Cumulative
Frequency
Distribution
1
Cumulative Frequency

0.8

0.6

0.4

0.2

0
76   80     84         88           92   96      100

FINANCE SCHOOL                         805A-44A-1033-VG-38
Symmetrical Frequency
Distribution

Frequency

50%   50%

Mean            X
Median
Mode

FINANCE SCHOOL           805A-44A-1033-VG-39
Skewed Frequency
Distributions
MODE

MEDIAN
F
MEAN

X
(1) SKEWED RIGHT                    MODE

F       MEDIAN
MEAN

X
(2) SKEWED LEFT
FINANCE SCHOOL               805A-44A-1033-VG-40
Descriptive Measures

SUMMARY
• Compute and explain the uses of the following
measures of central tendency and dispersion:

- Mean             - Range.
- Median           - Variance.
- Mode             - Standard Deviation.

• Construct and plot a frequency distribution and
identify the skewness of the distribution.

FINANCE SCHOOL              805A-44A-1033-VG-41
Terminal Learning Objective

Action: Perform descriptive measures statistical procedures
by computing the measures of central tendency and
dispersion; constructing a frequency distribution
(absolute, relative, and cumulative); plotting a
histogram, frequency polygon, and cumulative
frequency distribution; and identify the skewness of
the direction.

Conditions: Given the Formulas provided in the Summary Sheet
and a data set (population or sample).

Standard: IAW the Descriptive Measures Programmed Text and
Accepted Statistical Techniques

FINANCE SCHOOL                   805A-44A-1033-VG-42

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 views: 4 posted: 1/20/2012 language: pages: 42