# Average Salaries of Real Estate Agents in Missouri - PowerPoint by yin79391

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```									Describing Data:
Numerical Measures

Chapter 3
GOALS

1. Calculate the arithmetic mean, weighted mean,
median, mode, and geometric mean.
2. Explain the characteristics, uses, advantages, and
disadvantages of each measure of location.
3. Identify the position of the mean, median, and mode
for both symmetric and skewed distributions.
4. Compute and interpret the range, mean deviation,
variance, and standard deviation.
5. Understand the characteristics, uses, advantages,
and disadvantages of each measure of dispersion.
6. Understand Chebyshev’s theorem and the Empirical
Rule as they relate to a set of observations.
2
Characteristics of the Mean

The arithmetic mean is the most widely used
measure of location. It requires the interval
scale. Its major characteristics are:
– All values are used.
– It is unique.
– The sum of the deviations from the mean
is 0.
– It is calculated by summing the values
and dividing by the number of values.

3
Population Mean

For ungrouped data, the population mean is the
sum of all the population values divided by the
total number of population values:

4
Population Mean
For ungrouped data (data not in a frequency distribution)

Sum of all the values in the population
Population Mean   =
Number of values in the population

X

N
Define Variables & Symbols
µ    =   The population mean = “mu”
N    =   Total number of observations
X    =   A particular value
Σ    =   Indicates the operation of adding = “sigma”
ΣX   =   Sum of the X values                                  5
A Parameter is a measurable characteristic of a
population.

The Kiers                         56,000
family owns                       42,000
four cars. The
23,000
following is the
current mileage                   73,000
on each of the
four cars. The
mean mileage for
the cars is 40,333
1/3 miles.

6
EXAMPLE – Population Mean

7
Sample Mean

   For ungrouped data, the sample mean
is the sum of all the sample values
divided by the number of sample
values:

8
Sample Mean
For ungrouped data (data not in a frequency distribution)

Sum of all the values in the sample
Sample Mean        =
Number of values in the sample

X
X
n
Define Variables & Symbols
X
X    =   Sample Mean = "X bar"
n    =   Total number of observations
X    =   A particular value
Σ    =   Indicates the operation of adding = “sigma”
ΣX   =   Sum of the X values                               9
EXAMPLE – Sample Mean

10
A statistic is a measurable characteristic of a sample.

A sample of
five                       14.0,
executives                 15.0,
following
16.0,
bonus last
year (\$000):               15.0

Find The
Sample
Mean:                                             11
Properties of the Arithmetic Mean

   Every set of interval-level and ratio-level data has a mean.
   All the values are included in computing the mean.
   A set of data has a unique mean.
   The mean is affected by unusually large or small data values.
   The arithmetic mean is the only measure of central tendency
where the sum of the deviations of each value from the mean is
zero.

12
Sum Of The Deviations Of Each Value From
The Mean Is Zero
Deviation  ( X  X )
X  A particular value, X  Mean
Consider the set of values: 3, 8, and 4. The mean is 5.

( X  X )  (3  5)  (8  5)  (4  5)  0

Later, this will become relevant information when we calculate the variation and
13
standard deviation (This is the reason we will have to square the deviations).
Weighted Mean

   The weighted mean of a set of numbers X1,
X2, ..., Xn, with corresponding weights w1,
w2, ...,wn, is computed from the following
formula:

14
Weighted Mean
observation by the number of times it happens.
• The weighted mean of a set of numbers X1, X2, ..., Xn,
with corresponding weights w1, w2, ...,wn, is computed
from the following formula:

Xw 
(w 1X1  w 2 X 2  ...  w n X n )
Xw      
 (wX)
w
or
(w 1  w 2  ...w n )
Define Variables & Symbols
X w = Weighted Mean = "X bar sub w"
X1   =   A particular value
X2   =   A particular value
w1   =   A particular weight
w2   =   A particular weight
15
Σ    =   Indicates the operation of adding = “sigma”
EXAMPLE – Weighted Mean

The Carter Construction Company pays its hourly
employees \$16.50, \$19.00, or \$25.00 per hour.
There are 26 hourly employees, 14 of which are paid
at the \$16.50 rate, 10 at the \$19.00 rate, and 2 at the
\$25.00 rate. What is the mean hourly rate paid the
26 employees?

16
The Median

   The Median is the midpoint of the values after
they have been ordered from the smallest to
the largest.
–   There are as many values above the median as
below it in the data array.
–   For an odd set of values, the median will be the
middle number.
–   For an even set of values, the median will be the
arithmetic average of the two middle numbers.
–   Median is the measure of central tendency usually used by
real estate agents. Why?...

17
Median Example
\$80,000.00 \$70,000.00 \$275,000.00 \$65,000.00 \$60,000.00

Prices Ordered                  Prices Ordered
from Low to High                from High to Low
\$60,000                 \$275,000
65,000                80,000
70,000 ← Median →     70,000
2nd                  80,000                65,000
example:           275,000                 60,000
Prices Ordered                                 Prices Ordered
from Low to High                               from High to Low
\$60,000                                \$275,000
65,000                               80,000
70,000   (70000 + 75000)             75,000
= \$72,500
75,000          2                    70,000
80,000                               65,000
275,000                                60,000
18
Median Example in Excel

19
Properties of the Median

   There is a unique median for each data set.
   It is not affected by extremely large or small
values and is therefore a valuable measure
of central tendency when such values occur.
   It can be computed for ratio-level, interval-
level, and ordinal-level data.
   It can be computed for an open-ended
frequency distribution if the median does not
lie in an open-ended class.

20
EXAMPLES - Median

The ages for a sample of   The heights of four
five college students    basketball players, in
are:                     inches, are:
21, 25, 19, 20, 22             76, 73, 80, 75

Arranging the data in
Arranging the data in
ascending order gives:
ascending order gives:
73, 75, 76, 80.
19, 20, 21, 22, 25.
Thus the median is 75.5
Thus the median is 21.
21
Mode
•   The Mode is the value of the observation
that appears most frequently.
•   The mode is especially useful in
describing nominal and ordinal levels of
measurement.
•   There can be more than one mode.

22
Mode Example – Nominal Level
Data
Company has developed five bath oils
Company conducts marketing survey to determine which oil customers prefer
Number of Respondents Favoring Various Bath Oils

400
358
Number of Respondents

300

200                              193

110
100                                            92
43
0
Amor       Lamoure     Soothing    Smell Nice   Far Out

23
Mode       Bath Oil
Mode Example – Nominal Level
Data
• With Nominal Data      Respondents
Smell Nice     Frequency Table
you would count to     Lamoure       Smell Nice     92   =COUNTIF(\$A\$2:\$A\$797,C3)
see which occurs       Lamoure       Lamoure      358    =COUNTIF(\$A\$2:\$A\$797,C4)
Soothing      Soothing     193    =COUNTIF(\$A\$2:\$A\$797,C5)
most frequently        Smell Nice    Amor         110    =COUNTIF(\$A\$2:\$A\$797,C6)

• You can build a
Smell Nice    Far Out        43   =COUNTIF(\$A\$2:\$A\$797,C7)
Soothing
Frequency Table        Amor
Lamoure
using the              Soothing
COUNTIF function       Soothing
Smell Nice
in Excel.              Smell Nice
Lamoure
• The one that           Amor
Lamoure
occurs the most is
the Mode
More data…

24
Example – Mode – Ratio Level Data

Salaries
\$   35,000
\$   49,100
\$   60,000
\$   60,000
\$   40,000
\$   58,000
\$   60,000
\$   60,000
\$   40,000
\$   65,000
\$   55,000
\$   60,000
\$   71,400
\$   60,000
\$   55,000 In Excel:
\$   60,000 =MODE(G2:G16)

25
Mean, Median, Mode Using Excel
Table 2–4 in Chapter 2 shows the prices of the 80 vehicles sold last month at Whitner Autoplex in
Raytown, Missouri. Determine the mean and the median selling price. The mean and the median
selling prices are reported in the following Excel output. There are 80 vehicles in the study. So the
calculations with a calculator would be tedious and prone to error.

26
Mean, Median, Mode Using “Descriptive
Statistics” in Excel (From Textbook)

27
The Relative Positions of the Mean,
Median and the Mode

28
The Geometric Mean
   Useful in finding the average change of percentages, ratios, indexes, or growth
rates over time.
   It has a wide application in business and economics because we are often
interested in finding the percentage changes in sales, salaries, or economic
figures, such as the GDP, which compound or build on each other.
   The geometric mean will always be less than or equal to the arithmetic mean.
   The GM gives a more conservative figure that is not drawn up by large values
in the set.
   The geometric mean of a set of n positive numbers is defined as the nth root of
the product of n values.
   The formula for the geometric mean is written:

29
Geometric Mean
• The GM of a set of n positive numbers is defined
as the nth root of the product of n values. The
formula is either (both are true):

GM %  1  n ( X 1)( X 2)( X 3)...(Xn)
GM %  n ( X 1)( X 2)( X 3)...(Xn) 1
Define Variables & Symbols
GM       = Geometric Mean
X1       = A particular number (1 + %)
X2       = A particular number (1 + %)
n        = Number of postive numbers in set

30
Geometric Mean Example 1:
Percentage Increase

Starting Salary           \$41,000.00
Increase in salary Year 1     5%
Increase in salary Year 2    15%

GM  2 (1.05)(1.15)  1.09886

In Excel:
1.05 * 1.15 = 1.2075
GM = 1.2075 ^ (1/2) - 1 = 9.886%

31
Verify Geometric Mean Example
Verify 1:
Raise 1 = \$41,000.00 *       5% =    \$2,050.00
Raise 2 = 43,050.00 *       15% =     6,457.50
Total                                \$8,507.50

Verify 2:
Raise 1 = \$41,000.00 * 0.09886 =     \$4,053.39
Raise 2 = 45,053.39 * 0.09886 =       4,454.11
Total                                \$8,507.50

The GM gives a more
If We used Arithmetic Mean (5%+15%)/2 = 10%
conservative figure
that is not drawn     Raise 1 = \$41,000.00 *      10% = \$4,100.00
up by large values    Raise 2 = 45,100.00 *       10% =    4,510.00
in the set.           Total                              \$8,610.00
32
EXAMPLE – Geometric Mean (2)

The return on investment earned by Atkins
construction Company for four successive
years was: 30 percent, 20 percent, -40
percent, and 200 percent. What is the
geometric mean rate of return on investment?

GM  4 (1.3 )(1.2 )( 0.6 )( 3.0 )  4 2.808  1.294

33
Another Use Of GM:
Ave. % Increase Over Time
• Another use of the geometric mean is to
determine the percent increase in sales,
production or other business or economic series
from one time period to another
• Where n = number of periods

(Value at end of all the periods)
GM  n                                           1
(Value at beginning of all the periods)

34
Example for GM: Ave. % Increase Over
Time
• The total number of females enrolled in
American colleges increased from 755,000 in
1992 to 835,000 in 2000. That is, the geometric
mean rate of increase is 1.27%.

835,000
GM  8               1  .0127
755,000
•The annual rate of increase is 1.27%
•For the years 1992 through 2000, the rate of
female enrollment growth at American colleges
was 1.27% per year                                  35
Dispersion

Why Study Dispersion?
–   A measure of location, such as the mean or the median,
only describes the center of the data. It is valuable from
that standpoint, but it does not tell us anything about the
–   For example, if your nature guide told you that the river
across on foot without additional information? Probably not.
You would want to know something about the variation in
the depth.
–   A second reason for studying the dispersion in a set of data
is to compare the spread in two or more distributions.

36
Dictionary Definitions:
Dispersion, Variation, Deviation
• Dispersion
• The spatial property of being scattered about over
an area or volume
• The degree of scatter of data, usually about an
average value, such as the median or mean
• Variation
• The act of changing or altering something slightly but
noticeably from the norm or standard
• Deviation
– A variation that deviates from the standard or norm

Deviation  ( X  X )
37
Dispersion
1. How spread out is the data?
•   What is the average of all the deviations?
2. A small value for a measure of dispersion
indicates that the data are clustered around
the typical value (mean)
•   Mean can fairly represent the data
3. A large value for a measure of dispersion
indicates that the data are not clustered
around the typical value (mean)
•   Mean may not fairly represent the data

38
Samples of Dispersions

39
Measures of Dispersion

   Range

   Mean Deviation

   Variance and Standard
Deviation

40
Range
• Range = Highest Value – Lowest Value
• Excel = MAX – MIN functions
– The difference between the largest and the smallest
value

– It is easy to compute and understand
– Only two values are used in its calculation
– It is influenced by an extreme value
41
Range Example
Boomerangs made per day over ten day period
Boomerangs made per day over ten day period
?
?
? ?       ?
? ? ?        ? ?
52 54 56 58 60     62 64 66 68

X

?
?
?                 ?                 ?
?           ?                 ?                 ?           ?
40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80

X
Boomerangs made per day at Gel Boomerangs      42

• Mean Deviation measures the mean amount by
which the values in a population, or sample, vary
from their mean
 X X
MD 
n
Define Variables & Symbols
X
X    =   Sample Mean = "X bar"
n    =   Total number of observations
X    =   A particular value
MD   =   Mean Deviation
?    =   Absolute Value (distance from zero)   43
Variance
( X   )             2
( X  X )                    2
   2
                            s   2
N                              n 1
Define Variables & Symbols                    Define Variables & Symbols
 2 = Pop. Variance = "sigma squared"    s2       = Sample Variance
Total number of observations
n       =   (n as denominator tends to
N   = Total number of observations
underestimate variance)
Provides the appropriate correction
n-1 =
X   = A particular value                              for underestimation
X        =   A particular value
μ   = Pop. Mean                          X        =   Sample Mean

In Excel: 1) Population Variance  VARP function
2) Sample Variance  VAR function
44
Standard Deviation
( X   )                    2              ( X  X )2
          2                            s  s2 
N                                           n 1
Define Variables & Symbols                  Define Variables & Symbols

  2     = Pop. Standard Deviation        s  s2   = Sample Standard Deviation
Total number of observations
n      = (n as denominator tends to
N       = Total number of observations              underestimate variance)
Provides the appropriate correction
n-1     =
for underestimation
X        = A particular value               X      = A particular value
μ       = Pop. Mean                       X       = Sample Mean

In Excel: 1) Population Variance  STDEVP function
2) Sample Variance  STDEV function
The primary use of the statistic s2 is to estimate σ2, therefore
(n-1) is necessary to get a better representation of the
45
deviation or dispersion in the data (n tends to underestimate)
EXAMPLE – Range

The number of cappuccinos sold at the Starbucks location in the
Orange Country Airport between 4 and 7 p.m. for a sample of 5
days last year were 20, 40, 50, 60, and 80. Determine the mean
deviation for the number of cappuccinos sold.

Range = Largest – Smallest value
= 80 – 20 = 60

46
EXAMPLE – Mean Deviation

The number of cappuccinos sold at the Starbucks location in the
Orange Country Airport between 4 and 7 p.m. for a sample of 5
days last year were 20, 40, 50, 60, and 80. Determine the mean
deviation for the number of cappuccinos sold.

47
EXAMPLE – Variance and Standard
Deviation

The number of traffic citations issued during the last five months in
Beaufort County, South Carolina, is 38, 26, 13, 41, and 22. What
is the population variance?

2 ^(1/2) =  = 10.22441
48
EXAMPLE – Sample Variance and
Sample Standard Deviation

The hourly wages for
a sample of part-
time employees at
Home Depot are:
\$12, \$20, \$16, \$18,
and \$19. What is
the sample
variance and
sample standard
deviation?

s2 ^(1/2) = s = 3.162278
49
EXAMPLE – Sample Standard
Deviation (p1)

50
EXAMPLE – Sample Standard Deviation (p2)
Conclusions
The Variance is 14.48 square grams
The sample standard deviation is 3.8 grams
On average the weight of each box deviated from the sample mean by 3.8
grams. If the manufacturer of the cereal has 254 grams printed on the box, is
the production efficient? Is their claim of 254 grams accurate or fair?

If the historical sample standard deviation (s) is 2.7 grams, then an s = 3.8
indicates that it is reasonable to assume that the production is not efficient.
Also, s = 2.7 grams would indicate that the claim on the box does not seem
reasonable.

51
Chebyshev’s Theorem

The arithmetic mean biweekly amount contributed by the Dupree
Paint employees to the company’s profit-sharing plan is \$51.54,
and the standard deviation is \$7.51. At least what percent of the
contributions lie within plus 3.5 standard deviations and minus
3.5 standard deviations of the mean?

52
Chebyshev’s Theorem
• For any set of observations (sample or
population & does not have to be normal
distributed), the proportion of the values
that lie within (+/-) k standard deviations
of the mean is at least:

1
1
2) The smaller this
3) The
bigger this                2
k           1) The bigger this

• where k is any constant greater than 1          53
The Empirical Rule (also called Normal Rule)

54
34.13                       34.13
% area                      % area

13.58                                         13.58
% area                                        % area

2.15%                                                            2.15%
area                                                             area
0.13%
0.13%
area
area

Xbar-3   Xbar-2 Xbar-1      Xbar   Xbar +1 Xbar +2 Xbar +3

68.26% area

95.44% area

99.74% area
55
A sample of the rental rates at for 2 bedroom apartments in Seattle approximates
a symmetrical, bell-shaped distribution. The sample mean is \$925.00; The
standard deviation is \$110.00. Using the Empirical Rule, answer these questions:

Sample Mean = Xbar =                                                  \$925.00
Sample Standard Deviation = s =                                       \$110.00

Q1: About 68% of the rental rates are between what two amounts?
Xbar +/- 1*s = \$925.00 +/- 1*\$110.00 ==> About 68% of the values lie between
\$815.00 and \$1,035.00.
Q2: About 95% of the rental rates are between what two amounts?
Xbar +/- 2*s = \$925.00 +/- 2*\$110.00 ==> About 95% of the values lie between
\$705.00 and \$1,145.00.
Q3: Almost all of the rental rates are between what two amounts?
Xbar +/- 3*s = \$925.00 +/- 3*\$110.00 ==> About Almost all of the values lie
between \$595.00 and \$1,255.00.

56
The Arithmetic Mean of Grouped Data

57
The Arithmetic Mean of Grouped Data -
Example

Recall in Chapter 2, we
constructed a frequency
distribution for the vehicle
selling prices. The
information is repeated
below. Determine the
arithmetic mean vehicle
selling price.

58
The Arithmetic Mean of Grouped Data -
Example

59
Standard Deviation of Grouped Data

60
Standard Deviation of Grouped Data -
Example
Refer to the frequency distribution for the Whitner Autoplex data
used earlier. Compute the standard deviation of the vehicle
selling prices

61
Approximate Median from Frequency Distribution
1. Locate the class in which the median lies
• Divide the total number of data values by 2.
• Determine which class will contain this value. For
example, if n=50, 50/2 = 25, then determine which
class will contain the 25th value
n
2. Use formula to arrive                            CF
at estimate of Median:           Median  L  2      (i )
f
Define Variables & Symbols
L    =   Lower limit of the class containing the median
n    =   Total number of frequencies
f    =   Frequency in the median class
Cumulative number of frequencies in all the classes
CF   =
preceeding the class containing the medain
i    =   width of the class in which the median lies           62
Approximate Median from Frequency Distribution Example 1
Selling
Price               Cumulative         Total Number Frequencies
1st                              = 40
(\$       Frequency Frequency                     2
thousands)     (f)      CF
12 up to 15      8        8                40th value must be here
15 up to 18     23        31
18 up to 21     17        48       2nd         L            =         18
21 up to 24     18        66                   n            =         80
24 up to 27      8        74                   f            =         17
27 up to 30      4        78                  CF            =         31
30 up to 33      2        80
i            =          3
Total       80

n                  80
 CF                 31
Median  L  2      (i )  18  2       (3)  19.588235294
f                 17
The estimated median vehicle selling price is \$19,588.00
63
Selling
Price               Cumulative
(\$       Frequency Frequency
thousands )     (f)       CF
12 up to 15       8         8
15 up to 18      23        31
18 up to 21      17        48
21 up to 24      18        66
24 up to 27       8        74
27 up to 30       4        78
30 up to 33       2        80
Total        80
– n/2 – CF = 80/2-31 = 9 vehicles to get from the 31st to
the 40th
– (n/2 – CF)/f = 9/17 = 9/17th of the distance through the
class width of \$3000
– ((n/2 – CF)/f)(i) = (9/17)(\$3000) = \$1,588
– ((n/2 – CF)/f)(i) + L = \$1,588 + \$18,000 = \$19,588,
Median estimate
64
Estimate Range from
Grouped Data
Employee \$ contributed to profit   Frequency
sharing plan                    f
\$50      Up To        \$55             4
55      Up To        \$60             7
60      Up To         65             9
65      Up To         70            22
70      Up To         75            40
75      Up To         80            24
80      Up To         85            15
85      Up To         90             9
Total                     130

Upper Limit of        Lower Limit of
Estimated      the Highest           the Lowest
Range    =       Class           -     Class

Estimated
Range    =         \$90           -         \$50        = \$40

65
End of Chapter 3

66

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