QBM117 Business Statistics

1
QBM117Business Statistics

• Descriptive Statistics
• Numerical Descriptive Measures

2
Objectives

• To introduce numerical measures for describing
  the central location of data
• To introduce numerical measures for describing
  the variability of data

3
Numerical Descriptive Methods

• We have looked at tabular and graphical methods
  for presenting data.
• Although these methods help us to highlight
  important features of the data, they do not tell
  the whole story.
• Numerical descriptive measures allow us to be
  more precise in describing the characteristics of
  the data.

4
Numerical Descriptive Methods for Quantitative
Data

• Most numerical descriptive measures are obtained
  through arithmetic operations on the data.
• Arithmetic calculations can only be applied to
  quantitative data.
• Consequently most of the numerical descriptive
  measures we will discuss are for quantitative
  data.

5
Parameters and Statistics

• Recall the terms introduced in lecture 2 week 1
  population, sample, parameter, statistic
• Numerical measures calculated from sample data
  are called sample statistics.
• Numerical measures calculated from population
  data are called population parameters.

6

• We will look at a number of descriptive
  statistics and for each we will learn how to
  calculate both the population parameter and the
  sample statistic.
• In practice we usually collect data from a sample
  and calculate sample statistics to use as
  estimates of population parameters.

7
Notation

• Statistics are usually represented by Roman
  letters
• sample mean
• sample standard deviation s
• Parameters are usually represented by Greek
  letters
• population mean ?
• population standard deviation ?

8
Properties of numerical data

• Three major properties that describe quantitative
  data are
• - measures of central tendency
• - measures of dispersion
• - measures of shape

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Measures of Central Tendency

• In most sets of data there is a tendency for the
  data to group about a central point.
• This phenomenon is referred to as central
  tendency.
• We will look at three measures of central
  tendency mean, median and mode

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The Mean

• The most popular and useful measure of central
  tendency is the arithmetic mean, widely known as
  the average.
• The mean is calculated by summing all the
  observations and dividing by the number of
  observations.
• It can easily be calculated using the statistics
  function on your calculator.

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• The mean of a sample of n measurements
• is defined as
• The mean of a population of N measurements
• is defined as

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• I have shown you the formulas so that you
  understand how the mean is calculated.
• However it is expected that you will calculate
  the mean using the statistics function on your
  calculator.
• If you are unsure of how to use the statistics
  functions on your calculator refer to your
  calculator manual.
• The population mean or the sample mean are
  calculated using the same button on your
  calculator.

13
Example 1

• The following data are the price-earnings ratios
  for a set of stocks whose prices are quoted by
  NASDAQ
• Calculate the mean of the data.

4 20 16 28 31
10 23 37 29 15
33 21 18 35 29
14
The Median

• The median is the middle value when the data are
  arranged in order.
• To calculate the median
• - Order the data from smallest to largest
• - If the number of observations is odd, the
  median is the middle value.
• - If the number of observations in even, the
  median is the mean of the two middle
  observations.

15
Example 1 revisited

• The following data are the price-earnings ratios
  for a set of stocks whose prices are quoted by
  NASDAQ
• Calculate the median.

4 20 16 28 31
10 23 37 29 15
33 21 18 35 29
16

• Order the data.
• 4 10 15 16 18 20 21 23 28 29 29 31 33
  35 37
• median
• There are 15 observations and so the median will
  be the middle value.
• It will be the 8th value.

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Stem and Leaf Display

• A useful tool for ordering data is the stem and
  leaf display.
• To construct and stem and leaf display separate
  each observation into
• a stem, consisting of all but the last digit
• and a leaf, the final digit.
• Write the stems in a vertical column (smallest at
  top) .
• Write each leaf in the row to the right of the
  stem.
• Redraw, ordering the leaves.

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Example 1 revisited

• The following data are the price-earnings ratios
  for a set of stocks whose prices are quoted by
  NASDAQ
• Construct and stem and leaf display and
  calculate the median.

4 20 16 28 31
10 23 37 29 15
33 21 18 35 29
19
0 4
1 6 0 5 8
2 0 8 3 9 1 9
3 1 7 3 5

• Ordered

0 4
1 0 5 6 8
2 0 1 3 8 9 9
3 1 3 5 7
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The Mode

• The mode is the value that occurs most
  frequently.
• The mode doesnt necessarily lie in the middle.
• Its claim to be a measure of central tendency is
  based on the fact that it indicates the location
  of greatest concentration of values.
• The mode is a measure of central tendency that
  can be used for qualitative data.

21
Example 1 revisited

• The following data are the price-earnings ratios
  for a set of stocks whose prices are quoted by
  NASDAQ
• Calculate the mode.
• mode 29

4 20 16 28 31
10 23 37 29 15
33 21 18 35 29
22

• If no data value occurs more than once then there
  is no mode.
• A data set may have more than one mode.
• If there are two modes then the data are bimodal.
• If there are more than two modes the data are
  multimodal.

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Example 2

• A survey of television-viewing habits among
  university students provided the following data
  on viewing time in hours per week
• Calculate the mean, median and mode.

14 9 12 4 20 26 17 15
18 15 10 6 16 15 8 5
24

• mean 13.125
• 4 5 6 8 9 10 12 14 15 15 15 16 17
  18 20 26
• median
• median 14.5
• mode 15

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Mean, Median or Mode

• There are several factors to consider when making
  our choice of measure of central tendency.
• The mean is generally our first selection.
• However, there are circumstances when the median
  is better.
• The mode is seldom the best measure of central
  tendency.

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• The mean is a popular measure because it is
  simple to calculate and interpret, and lends
  itself to mathematical manipulation.
• However the mean is sensitive to skewness and
  outliers.
• The mean can be thought of as the balance point
  of the data.
• If there are a few data points that are far from
  the bulk of the data, the mean moves towards them
  in order to maintain balance.

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• The mean is the preferred measure of central
  tendency.
• However, if the data are skewed or contain
  outliers then the median is the preferred measure
  of central tendency.
• If the data are qualitative, the mode must be
  used.

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Relationship between Mean, Median and Mode

• If the data is unimodal and symmetric, the mean,
  median and mode coincide.
• If the data are unimodal and positively skewed,
  the mean is greater than the median, which is
  greater than the mode.
• If the data are unimodal and negatively skewed,
  the mean is less than the median, which is less
  than the mode.

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Measures of Dispersion

• In addition to knowing the central location of
  the data values, it is important to know how the
  values vary about this point.
• We are now going to look at measures of
  dispersion, also referred to as
• - measures of spread
• - measures of variability
• We will look at three measures of dispersion
• range, standard deviation and coefficient of
  variation

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The Range

• The range is the difference between the largest
  and smallest observations in a data set.
• The range measures the total spread of the data
  set.
• Although the range is a simple measure of
  variability, it does not take into account how
  the data are distributed between the smallest and
  largest values.
• Hence the range is seldom used as the only
  measure.

31
Example 1 revisited

• The following data are the price-earnings ratios
  for a set of stocks whose prices are quoted by
  NASDAQ
• Calculate the range.
• range 37 4 33

4 20 16 28 31
10 23 37 29 15
33 21 18 35 29
32
Variance and Standard Deviation

• The variance and the standard deviation are the
  two most widely accepted measures of dispersion.
• The variance is the square root of the standard
  deviation.
• Both measures take into account how far each data
  value is away from the mean.

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Population Variance

• The variance of a population of N measurements
• having mean ? is defined as

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Sample Variance

• The variance of a sample of n measurements
• having mean is defined as

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Standard Deviation

• Calculating the variance involves squaring the
  original measurements and hence the unit attached
  to the variance is the square of the unit
  attached to the original measurements.
• Taking the square root of the variance gives as a
  measure of variability that is in the same units
  as the data.
• This measure is the standard deviation.

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Population Standard Deviation

• The standard deviation of a population of N
  measurements having mean µ is defined as

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Sample Standard Deviation

• The standard deviation of a sample of n
  measurements having mean is defined as

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Calculating the Standard Deviation and Variance

• As with the mean, you are expected to calculate
  the standard deviation and variance using the
  statistics functions on your calculator.
• You are not to use the formulae, these have been
  provided to help you understand what the standard
  deviation and variance are.
• Note that the population standard deviation and
  sample standard deviation are calculated using
  different buttons on your calculator.

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Important Points about the Standard Deviation

• The standard deviation cannot be negative.
• The standard deviation is zero if, and only if,
  all of the observations have the same value.
• Like the mean, the standard deviation is not
  resistant. Strong skewness or a few outliers can
  greatly increase the standard deviation.

40
Example 1 revisited

• The following data are the price-earnings ratios
  for a set of stocks whose prices are quoted by
  NASDAQ
• Calculate the standard deviation and the
  variance.

4 20 16 28 31
10 23 37 29 15
33 21 18 35 29
41
Coefficient of Variation

• In some situations we may be interested in a
  measure of variability that indicates how large
  the standard deviation is in relation to the
  mean.
• This measure is called the coefficient of
  variation (CV) and is calculated by dividing the
  standard deviation of a data set by the mean.
• The CV allows us to compare the variability of
  two data sets having different units of
  measurement.

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• A standard deviation of 1mm would be considered
  very large for the measured thickness of CDs on a
  production line.
• However a standard deviation of 1mm would be
  considered small for the height of a telephone
  pole.
• When the means for data sets differ greatly we do
  not get an accurate picture of the relative
  variability in the two data sets by comparing the
  standard deviations.

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Calculating the Coefficient of Variation

• The sample coefficient of variation is calculated
  by
• The population coefficient of variation is
  calculated by

44
Example 1 revisited

• The following data are the price-earnings ratios
  for a set of stocks whose prices are quoted by
  NASDAQ
• Calculate the coefficient of variation.

4 20 16 28 31
10 23 37 29 15
33 21 18 35 29
45
Example 2 revisited

• A survey of television-viewing habits among
  university students provided the following data
  on viewing time in hours per week
• Calculate the range, standard deviation,
  variance and coefficient of variation.

14 9 12 4 20 26 17 15
18 15 10 6 16 15 8 5
46

• range 26 6 20
• standard deviation s 5.92 (2d.p.)
• variance s2 35.05
• coefficient of variation cv 0.45 (2d.p.)

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Interpreting the Standard Deviation

• The standard deviation, as a measure of average
  deviation around the mean, helps you understand
  how the observations are distributed above and
  below the mean.
• A data set with a large standard deviation has
  much dispersion with values widely scattered
  around its mean.
• A data set with a small standard deviation has
  little dispersion with the values tightly
  clustered about the mean.

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Chebyshes Theorem

• More than a century ago, Russian mathematician
  Pavroty Chebyshev, found that regardless of how a
  data set is distributed, the proportion of
  observations that are contained within distances
  of k standard deviations of the mean is at least
  1-(1/k2).
• This is known as Chebyshevs theorem.

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• Regardless of the shape of the distribution,
  Chebyshevs theorem states
• At least 75 of the observations must lie within
  2 standard deviations of the mean
• At least 89 of the observations must lie within
  3 standard deviations of the mean
• At least 94 of the observations must lie within
  4 standard deviations of the mean

50
Example 3.11 from text (pg 86)

• The duration (in minutes) of a sample of 30
  long-distance telephone calls placed by a firm in
  Melbourne in a given week are given in Table 3.2
  on page 86 of the text.
• The 30 telephone-call durations have a mean of
  10.26 and a standard deviation of 4.29.
• Chebyshevs theorem states that at least 75 of
  the call durations lie within 2 standard
  deviations of the mean.

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• When we look at the data we find that all but
  the largest of the 30 durations fall within this
  interval.
• That is, the interval actually contains 96.7 of
  the call durations.

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Empirical Rule

• A more exact rule applies if the distribution of
  the data is bell-shaped.
• The empirical rule has evolved from empirical
  studies that have produced samples possessing
  bell-shaped distributions.

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• The empirical rule states that for data with a
  bell-shaped distribution
• About 68 of all observations lie within 1
  standard deviation of the mean
• About 95 of all observations lie within 2
  standard deviations of the mean
• Almost all 94 of the observations lie within 3
  standard deviations of the mean

54
Example 3.12 from text (pg 87)

• The data in the sample of telephone-call
  durations in Table 3.2 have a mean of 10.26, a
  standard deviation of 4.29, and the durations
  have an approximately bell-shaped distribution
  (see Figure 3.5).
• According to the empirical rule, approximately
  68 of the observations should lie in the
  interval

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• According to the empirical rule, approximately
  68 of the observations should lie in the
  interval
• If we look at the data we see that 21 out of the
  30 durations are contained in this interval, i.e.
  70.
• This is very close the the empirical rules
  approximation.

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• According to the empirical rule, approximately
  95 of the observations should lie in the
  interval
• If we look at the data we see that 29 out of the
  30 durations are contained in this interval, i.e.
  96.7.
• This is very close the the empirical rules
  approximation.

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• Reading for next lecture
• Chapter 3 Sections 3.5 - 3.6
• Exercises
• 3.7
• 3.20
• 3.25a
• 3.31