Describing Data: Numerical Measures

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Describing DataNumerical Measures

• Chapter 3

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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 Chebyshevs theorem and the Empirical
   Rule as they relate to a set of observations.

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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.

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Population Mean

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

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Population MeanFor ungrouped data (data not in a
frequency distribution)
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A Parameter is a measurable characteristic of a
population.
56,000
The Kiers family owns four cars. The following
is the current mileage on each of the four cars.
The mean mileage for the cars is 40,333 1/3
miles.
42,000
23,000
73,000
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EXAMPLE Population Mean
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Sample Mean

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

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Sample MeanFor ungrouped data (data not in a
frequency distribution)
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EXAMPLE Sample Mean
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A statistic is a measurable characteristic of a
sample.
A sample of five executives received the
following bonus last year (000)
14.0, 15.0, 17.0, 16.0, 15.0
Find The Sample Mean
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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.

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Sum Of The Deviations Of Each Value From The Mean
Is Zero
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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

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Weighted Mean

• Instead of adding all the observations, we
  multiply each 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

or
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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?
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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?...

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Median Example
2nd example
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Median Example in Excel
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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.

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EXAMPLES - Median

• The ages for a sample of five college students
  are
• 21, 25, 19, 20, 22
• Arranging the data in ascending order gives
• 19, 20, 21, 22, 25.
• Thus the median is 21.

The heights of four basketball players, in
inches, are 76, 73, 80, 75 Arranging the data
in ascending order gives 73, 75, 76, 80.
Thus the median is 75.5
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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.

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Mode Example Nominal Level Data
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Mode Example Nominal Level Data

• With Nominal Data you would count to see which
  occurs most frequently
• You can build a Frequency Table using the COUNTIF
  function in Excel.
• The one that occurs the most is the Mode

More data
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Example Mode Ratio Level Data
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Mean, Median, Mode Using Excel
Table 24 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.
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Mean, Median, Mode Using Descriptive Statistics
in Excel (From Textbook)
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The Relative Positions of the Mean, Median and
the Mode
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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

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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)

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Geometric Mean Example 1Percentage Increase
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Verify Geometric Mean Example
The GM gives a more conservative figure that is
not drawn up by large values in the set.
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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?

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Another Use Of GMAve. 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

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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.

• 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

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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 spread of the
  data.
• For example, if your nature guide told you that
  the river ahead averaged 3 feet in depth, would
  you want to wade 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.

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Dictionary DefinitionsDispersion, 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

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Dispersion

• How spread out is the data?
• What is the average of all the deviations?
• 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
• 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

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Samples of Dispersions
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Measures of Dispersion

• Range
• Mean Deviation
• Variance and Standard Deviation

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Range

• Range Highest Value Lowest Value
• Excel MAX MIN functions
• The difference between the largest and the
  smallest value
• Advantage
• It is easy to compute and understand
• Disadvantage
• Only two values are used in its calculation
• It is influenced by an extreme value

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Range Example
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Mean Deviation (MA or MAD)

• Mean Deviation measures the mean amount by which
  the values in a population, or sample, vary from
  their mean

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Variance
In Excel 1) Population Variance ? VARP function
2) Sample Variance ? VAR function
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Standard Deviation
In Excel 1) Population Variance ? STDEVP
function 2) Sample Variance ? STDEV
function
The primary use of the statistic s2 is to
estimate s2, therefore (n-1) is necessary to get
a better representation of the deviation or
dispersion in the data (n tends to underestimate)
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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
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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.

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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
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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
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EXAMPLE Sample Standard Deviation (p1)
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EXAMPLE Sample Standard Deviation
(p2)Conclusions
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Chebyshevs Theorem

• The arithmetic mean biweekly amount contributed
  by the Dupree Paint employees to the companys
  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?

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Chebyshevs 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
• where k is any constant greater than 1

2) The smaller this
3) The bigger this
1) The bigger this
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The Empirical Rule (also called Normal Rule)
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The Arithmetic Mean of Grouped Data
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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.

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The Arithmetic Mean of Grouped Data - Example
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Standard Deviation of Grouped Data
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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

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Approximate Median from Frequency Distribution

• 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 n50, 50/2 25, then determine
  which class will contain the 25th value

1. Use formula to arrive at estimate of Median

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Approximate Median from Frequency Distribution
Example 1
40th value must be here
The estimated median vehicle selling price is
19,588.00
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• Lets think about this
• 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

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Estimate Range from Grouped Data
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End of Chapter 3