Business Statistics for Managerial Decision Making

1
Business Statistics for Managerial Decision
Making

• Examining Distributions

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Introduction

• Descriptive Statistics
• Methods that organize and summarize data aid in
  effective presentation and increased
  understanding.
• Bar charts, tabular displays, various plots of
  economic data, averages and percentages.
• Often the individuals or objects studied by an
  investigator come from a much larger collection,
  and the researchers interest goes beyond just
  data summarization.

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Introduction

• Population
• The entire collection of individuals or objects
  about which information is desired.
• Sample
• A subset of the population selected in some
  prescribed manner for study.

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Introduction

• Inferential Statistics
• Involves generalizing from a sample to the
  population from which it was selected.
• This type of generalization involves some risk,
  since a conclusion about the population will be
  reached based on the basis of available, but
  incomplete, information.
• An important aspect in the development of
  inference techniques involves quantifying the
  associated risks.

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Individuals and variables

• Individuals
• are the objects described by a set of data.
• They may be people, but they may also be
  business firms, common stocks, or other objects.
• A Variable
• is any characteristic of an individual.
• A variable can take different values for
  different individuals.

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Categorical Quantitative Variables

• A Categorical Variable places an individual into
  one of several groups or categories.
• A Quantitative Variable takes numerical values
  for which arithmetic operations such as adding
  and averaging make sense.
• The distribution of a variable tell us what
  values it takes and how often it takes these
  values.

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Example
8
Example
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Discrete and Continuous Variable

• With numerical data (quantitative variables), it
  is useful to make a further distinction.
• Numerical data is discrete if the possible values
  are isolated points on the number line.
• Numerical data is continuous if the set of
  possible values form an entire interval on the
  number line.

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Stem plot

• To make a stem plot
• Separate each observation into a stem consisting
  of all but the final (rightmost) digit and a
  leaf, the final digit. Stems may have as many
  digits as needed, but each leaf contains only a
  single digit.
• Write the stems in a vertical column with the
  smallest at the top, and draw a vertical line at
  the right of this column.
• Write each leaf in the row to the right of its
  stem, in increasing order out from the stem.

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Stem plot
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Frequency Distribution

• A frequency distribution for categorical data is
  a table that displays the categories,
  frequencies, and relative frequencies.
• Example
• The increasing emphasis on exercise has resulted
  in an increase of sport related injuries. A
  listing of the 82 sample observations would look
  something like this
• F, Sp, Sp, Co, F, L, F, Ch, De, L, Sp, Di, St,
  Cn,

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Frequency Distribution

• The following coding is used
• Sp Sprain, St Strain, Di dislocation,
• Co Contusion, L laceration,
• Cn Concussion, F fracture,
• Ch chronic, De dental

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Frequency Distribution
15
Bar Graph
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Pie Chart
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Frequency Distribution for Discrete Numerical
Data

• Discrete numerical data almost always results
  from counting.
• In such cases, each observation is a whole
  number.
• For example, if the possible values are 0, 1, 2,
  3, , then these are listed in column, and a
  running tally is kept as a single pass is made
  through the data

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Frequency Distribution for Discrete Numerical Data

• Example
• A sample of 708 bus drivers employed by public
  corporations was selected, and the number of
  traffic accidents in which each was involved
  during a 4-year period was determined. A listing
  of the 708 sample observations would look
  something like this
• 3, 0, 6, 0, 0, 2, 1, 4, 1,

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Frequency Distribution for Discrete Numerical Data
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Bar Graph
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Frequency Distributions for Continuous Data

• The difficulty with continuous data, such as
  observations on the unemployment rate by state,
  is that there is no natural categories.
• Therefore we define our own categories. by
  marking off some intervals on horizontal
  unemployment rate axis as picture below.
• 1.00 9.00

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Frequency Distributions for Continuous Data

• If the smallest rate were 1.5, and the largest
  was 8.9, we might use the intervals of width 1
  with the first one starting at 1 and the last one
  ending at 9.
• Each data value should fall in exactly one of
  these intervals.

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Frequency Distributions for Continuous Data
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Frequency Distributions for Continuous Data
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Histograms

• Mark the boundaries of the class intervals on a
  horizontal axis.
• Draw a vertical scale marked with either relative
  frequencies or frequencies.
• The rectangle corresponding to a particular
  interval is drawn directly above the interval.
• The height of each rectangle is then the class
  frequency or relative frequency.

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Histograms
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Histograms
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Examining a Distribution

• In any graph of data, look for overall pattern
  and for striking deviation from that pattern.
• You can describe the overall pattern of a
  histogram by its shape, center, and spread.
• An important kind of deviation is an outlier, an
  individual value that falls outside the overall
  pattern.

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Symmetric Skewed Distribution

• A distribution is symmetric if the right and left
  sides of the histogram are approximately mirror
  images of each other.
• A distribution is skewed to the right if the
  right side of the histogram ( containing the half
  of the observations with larger values) extends
  much farther out than the left side.
• It is skewed to the left if the left side of the
  histogram extends much farther out than the right
  side.

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Symmetric Distribution
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Skewed to the Right
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Symmetric Distribution
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Numerical Summary Measures

• Describing the center of a data set.
• Mean
• Median
• Describing the variability in a data set.
• Variance, standard deviation
• Quartiles

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The Mean

• To find the mean of a set of observations, add
  their values and divide by the number of
  observations. If the n observations are
  , their mean is
• In a more compact notation,

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The Median

• The Median M is the midpoint of a distribution,
  the number such that half of the observations are
  smaller and the other half are larger. To find
  the median of a distribution
• Arrange all observations in order of size, from
  smallest to largest.
• If the number of observations n is odd, the
  median M is the center observation in the ordered
  list.
• If the number of observations n is even, the
  median M is the mean of the two center
  observations in the ordered list.

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The Quartiles Q1 and Q3

• To calculate the quartiles
• Arrange the observations in increasing order and
  locate the median M in the ordered list of
  observations.
• The first quartile Q1 is the median of the
  observations whose position in the ordered list
  is to the left of the location of the overall
  median.
• The third quartile Q3 is the median of the
  observations whose position in the ordered list
  is to the right of the location of the overall
  median.

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The Five Number Summary and Box-Plot

• The five number summary of a distribution
  consists of the smallest observation, the first
  quartile, the median, the third quartile, and the
  largest observation, written in order from
  smallest to largest. In symbols, the five number
  summary is
• Minimum Q1 M Q3 Maximum

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The Five Number Summary and Box-Plot

• A box-plot is a graph of the five number Summary.
• A central box spans the quartiles.
• A line in the box marks the median.
• Lines extend from the box out to the smallest and
  largest observations.
• Box-plots are most useful for side-by-side
  comparison of several distributions.

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Example
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The Standard Deviation s

• The Variance s2 of a set of observations is the
  average of the squares of the deviations of the
  observations from their mean. In symbols, the
  variance of n observations is
• or, more compactly,

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The Standard Deviation s

• The standard deviation s is the square root of
  the variance s2

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Choosing a Summary

• The five number summary is usually better than
  the mean and standard deviation for describing a
  skewed distribution or a distribution with
  extreme outliers. Use ,
• and s only for reasonably symmetric
  distributions that are free of outliers.

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Strategies for Exploring Data

• Plot the data
• Make a graph, usually a histogram or a stem-plot.
• Look at the distribution of the variable for
• overall pattern (shape, center, spread).
• striking deviations such as outliers.
• Calculate a numerical summary to briefly describe
  center and spread.
• Describe the overall pattern with a smooth curve.

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Density Curves

• Sometimes the overall pattern (the distribution
  of the variable) of a large number of
  observations is so regular that we can describe
  it by a smooth curve, called Density curve.
• The curve is a mathematical model for the
  distribution.

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Density Curve

• Histogram of the city gas mileage (miles per
  gallon) of 856, 2001 model year motor vehicle.
• The smooth curve, density curve, shows the
  overall shape of the distribution.

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Density Curve

• The proportion of cars with gas mileage less
  than 20 from the histogram is

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Density Curve

• The proportion of cars with gas mileage less than
  20 from the density curve is .410
• The area under the density curve gives a good
  approximation of areas given by histogram.

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Density Curve

• A density curve is a curve that
• Is always on or above the horizontal axis.
• Has area exactly 1 underneath it.
• A density curve describes the overall pattern of
  a distribution.
• The area under the curve and above any range of
  values is the proportion of all observations that
  fall in that range.

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Median and mean of a Density Curve

• The median of a density curve is the point that
  divides the area under the curve in Half.

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Median and Mean of a Density Curve

• The mean of a density curve is the balance point,
  at which the curve would balance if made of solid
  material.

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Median and Mean of a Density Curve

• The median and mean are the same for a symmetric
  density curve.
• They both are at the center of the curve.

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Median and Mean of a Density Curve

• The mean of a skewed curve is pulled away from
  the median in the direction of the long tail.

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Normal Density Curve

• These density curves, called normal curves, are
• Symmetric
• Single peaked
• Bell shaped
• Normal curves describe normal distributions.

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Normal Density Curve

• The exact density curve for a particular normal
  distribution is described by giving its mean ?
  and its standard deviation ?.
• The mean is located at the center of the
  symmetric curve and it is the same as the median.
• The standard deviation ? controls the spread of a
  normal curve.

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Normal Density Curve
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The 68-95-99.7 Rule

• Although there are many normal curve, They all
  have common properties. In particular, all Normal
  distributions obey the following rule.
• In a normal distribution with mean ? and standard
  deviation ?
• 68 of the observations fall within ? of the mean
  ?.
• 95 of the observations fall within 2? of ?.
• 99.7 of the observations fall within 3? of ?.

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The 68-95-99.7 Rule
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The 68-95-99.7 Rule
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Standard Normal Distribution

• The standard Normal distribution is the Normal
  distribution N(0, 1) with mean
• ? 0 and standard deviation ? 1.

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The standard Normal Table

• What is the area under the standard normal curve
  between z 0 and z 2.3?
• Compact notation
• P .9893 - .5 .4893

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Finding the area under a normal curve

1. State the problem in terms of the observed
   variable x.
2. Standardize x to restate the problem in terms of
   a standard normal variable z
3. Draw a picture to show the area under the
   standard Normal curve.
4. Find the required area under the standard Normal
   curve Using table A and the fact that the total
   area under the curve is 1.

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Example

• The annual rate of return on stock indexes
  (which combine many individual stocks) is
  approximately Normal. Since 1954, the Standard
  Poors 500 stock index has had a mean yearly
  return of about 12, with standard deviation of
  16.5. Take this Normal distribution to be the
  distribution of yearly returns over a long
  period. The market is down for the year if the
  return on the index is less than zero. In what
  proportion of years is the market down?

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Example

• State the problem
• Call the annual rate of return for Standard
  Poors 500-stocks Index x. The variable x has the
  N(12, 16.5) distribution. We want the proportion
  of years with
• X lt 0.
• Standardize
• Subtract the mean, then divide by the standard
  deviation, to turn x into a standard Normal z

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Example

• Draw a picture to show the standard normal curve
  with the area of interest shaded.
• Use the table
• The proportion of observations less than
• - 0.73 is .2327.
• The market is down on an annual basis about
  23.27 of the time.

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Example

• What percent of years have annual return between
  12 and 50?
• State the problem
• Standardize

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Example

• Draw a picture.
• Use table.
• The area between 0 and 2.30 is the area below
  2.30 minus the area below 0.
• 0.9893- .50 .4893

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Finding a Value when Given a Proportion

• Sometimes we may want to find the observed value
  with a given proportion of observations above or
  below it.
• To do this, use table A backward. Find the given
  proportion in the body of the table, read the
  corresponding z from the left column and top row,
  then unstandardize to get the observed value.

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Example

• Miles per gallon ratings of compact cars (2001
  model year) follow approximately the N(25.7,
  5.88) distribution. How many miles per gallon
  must a vehicle get to place in the top 10 of all
  2001 model year compact cars?

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Example

• We want to find the miles per gallon rating x
  with area 0.1 to its right under the Normal Curve
  with mean 25.7 and standard deviation 5.88. That
  is the same as finding the miles per gallon
  rating x with area 0.9 to its left.

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Example

• Look in the body of Table A for the entry closest
  to 0.9. It is 0.8997. This is the entry
  corresponding to z 1.28.

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Example

• Unstandardize to transform the solution from the
  z back to the original x scale.

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Standard Normal Distribution

• If a variable x has any normal distribution N(?,
  ?) with mean ? and standard deviation ?, then the
  standardized variable
• has the standard Normal distribution.
• This standardized value is often called z-score.

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The standard Normal Table

• Table A is a table of area under the standard
  Normal curve. The table entry for each value z is
  the area under the curve to the left of z.
• Or you can use the applet at the following site.
• http/www.stat.sc.eduwest/applets/normaldemo.html
  

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The standard Normal Table

• What is the area under the standard normal curve
  to the right of
• z - 2.15?
• Compact notation
• P 1 - .0158 .9842