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Introduction and Descriptive Statistics


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Title: Introduction and Descriptive Statistics

  • Introduction and Descriptive Statistics

STATISTICS The science of collecting, organizing,
presenting, analyzing, and interpreting data to
assist in making more effective decisions.
  • Statistics is a science that helps us make better
    decisions in business, economics and finance as
    well as in other fields.
  • Statistics teaches us how to summarize, analyze,
    and draw meaningful inferences from data that
    then lead to improve decisions.
  • These decisions that we make help us improve the
    running, for example, a department, a company,
    the entire economy, etc.

Using Statistics (Two Categories)
  • Descriptive Statistics
  • Collect
  • Organize
  • Summarize
  • Display
  • Analyze
  • Inferential Statistics
  • Predict and forecast
  • value of population parameters
  • Test hypothesis about value of population
    parameter based on
  • sample statistic
  • Make decisions

Samples and Populations
  • A population consists of the set of all
    measurements for which the investigator is
  • A sample is a subset of the measurements selected
    from the population.
  • A census is a complete enumeration of every item
    in a population.

Why Sample?
  • Census of a population may be
  • Impossible
  • Impractical
  • Too costly

Parameter Versus Statistic
Types of Data - Two Types
  • Qualitative - Categorical or Nominal
  • Examples are-
  • Color
  • Gender
  • Nationality
  • Quantitative - Measurable or Countable
  • Examples are-
  • Temperatures
  • Salaries
  • Number of points scored on a 100 point exam

Scales of Measurement
  • Nominal Scale - groups or classes
  • Gender
  • Ordinal Scale - order matters
  • Ranks (top ten videos)
  • Interval Scale - difference or distance matters
    has arbitrary zero value.
  • Temperatures (0F, 0C), Likert Scale
  • Ratio Scale - Ratio matters has a natural zero
  • Salaries


Population Mean
  • For ungrouped data, the population mean is the
    sum of all the population values divided by the
    total number of population values. The sample
    mean is the sum of all the sample values divided
    by the total number of sample values.

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.

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
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 Mode
Measures of 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
  • 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
  • A second reason for studying the dispersion in a
    set of data is to compare the spread in two or
    more distributions.

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.
  • Step 1 Compute the mean
  • Step 2 Subtract the mean (50) from each of the
    observations, convert to positive if difference
    is negative
  • Step 3 Sum the absolute differences found in
    step 2 then divide by the number of observations

Variance and Standard Deviation
  • The variance and standard deviations are
    nonnegative and are zero only if all observations
    are the same.
  • For populations whose values are near the mean,
    the variance and standard deviation will be
  • For populations whose values are dispersed from
    the mean, the population variance and standard
    deviation will be large.
  • The variance overcomes the weakness of the range
    by using all the values in the population

EXAMPLE Population Variance and Population
Standard Deviation
  • The number of traffic citations issued during the
    last twelve months in Beaufort County, South
    Carolina, is reported below
  • What is the population variance?
  • Step 1 Find the mean.
  • Step 2 Find the difference between each
    observation and the mean, and square that
  • Step 3 Sum all the squared differences found in
    step 3
  • Step 4 Divide the sum of the squared differences
    by the number of items in the population.

Sample Variance and Standard Deviation
EXAMPLE 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?
The Arithmetic Mean and Standard Deviation of
Grouped Data
EXAMPLE Compute the standard deviation of the
vehicle selling prices in the frequency table
  • Determine the arithmetic mean vehicle selling
    price given in the frequency table below.

Group Data and the Histogram
  • Dividing data into groups or classes or intervals
  • Groups should be
  • Mutually exclusive
  • Not overlapping - every observation is assigned
    to only one group
  • Exhaustive
  • Every observation is assigned to a group
  • Equal-width (if possible)
  • First or last group may be open-ended

Frequency Distribution
  • Table with two columns listing
  • Each and every group or class or interval of
  • Associated frequency of each group
  • Number of observations assigned to each group
  • Sum of frequencies is number of observations
  • N for population
  • n for sample
  • Class midpoint is the middle value of a group or
    class or interval
  • Relative frequency is the percentage of total
    observations in each class
  • Sum of relative frequencies 1

Example Frequency Distribution
x f(x) f(x)/n Spending Class () Frequency
(number of customers) Relative Frequency
0 to less than 100 30 0.163 100 to less than
200 38 0.207 200 to less than
300 50 0.272 300 to less than
400 31 0.168 400 to less than
500 22 0.120 500 to less than
600 13 0.070
184 1.000
  • Example of relative frequency 30/184 0.163
  • Sum of relative frequencies 1

Cumulative Frequency Distribution
x F(x) F(x)/n Spending Class
() Cumulative Frequency Cumulative Relative
Frequency 0 to less than 100 30
0.163 100 to less than 200 68 0.370 200
to less than 300 118 0.641 300 to less
than 400 149 0.810 400 to less than
500 171 0.929 500 to less than
600 184 1.000
The cumulative frequency of each group is the sum
of the frequencies of that and all preceding
  • A histogram is a chart made of bars of different
  • Widths and locations of bars correspond to widths
    and locations of data groupings
  • Heights of bars correspond to frequencies or
    relative frequencies of data groupings

Histogram Example
Relative Frequency Histogram
Frequency Histogram
Chebyshevs Theorem and Empirical Rule
Quartiles, Deciles and Percentiles
  • The standard deviation is the most widely used
    measure of dispersion.
  • Alternative ways of describing spread of data
    include determining the location of values that
    divide a set of observations into equal parts.
  • These measures include quartiles, deciles, and
  • To formalize the computational procedure, let Lp
    refer to the location of a desired percentile. So
    if we wanted to find the 33rd percentile we would
    use L33 and if we wanted the median, the 50th
    percentile, then L50.
  • The number of observations is n, so if we want to
    locate the median, its position is at (n 1)/2,
    or we could write this as (n 1)(P/100), where
    P is the desired percentile

Percentiles - Example
  • Listed below are the commissions earned last
    month by a sample of 15 brokers at Salomon Smith
    Barneys Oakland, California, office.
  • 2,038 1,758 1,721 1,637 2,097 2,047
    2,205 1,787 2,287 1,940 2,311 2,054
    2,406 1,471 1,460
  • Locate the median, the first quartile, and the
    third quartile for the commissions earned.
  • Step 1 Organize the data from lowest to largest
  • 1,460 1,471 1,637 1,721 1,758 1,787 1,9
    40 2,038 2,047 2,054 2,097 2,205 2,287
    2,311 2,406
  • Step 2 Compute the first and third quartiles.
    Locate L25 and L75 using

Measures of Variability or Dispersion
  • Range
  • Difference between maximum and minimum values
  • Interquartile Range
  • Difference between third and first quartile (Q3
    - Q1)
  • Variance
  • Averageof the squared deviations from the mean
  • Standard Deviation
  • Square root of the variance

??Definitions of population variance and sample
variance differ slightly.
  • Another characteristic of a set of data is the
  • There are four shapes commonly observed
    symmetric, positively skewed, negatively skewed,
  • The coefficient of skewness can range from -3 up
    to 3.
  • A value near -3, indicates considerable negative
  • A value such as 1.63 indicates moderate positive
  • A value of 0, which will occur when the mean and
    median are equal, indicates the distribution is
    symmetrical and that there is no skewness

The Relative Positions of the Mean, Median and
the Mode
Methods of Displaying Data
  • Pie Charts
  • Categories represented as percentages of total
  • Bar Graphs
  • Heights of rectangles represent group frequencies
  • Frequency Polygons
  • Height of line represents frequency
  • Ogives
  • Height of line represents cumulative frequency
  • Time Series Plots
  • Represents values over time
  • Stem-and-Leaf Displays
  • Quick listing of all observations
  • Conveys some of the same information as a
  • Box Plots
  • Median
  • Lower and upper quartiles
  • Maximum and minimum

Pie Chart
Bar Chart

Frequency Polygon and Ogive
(Cumulative frequency or relative frequency
Time Series Plot
Stem-and-Leaf Display
  • Stem-and-leaf display is a statistical technique
    to present a set of data. Each numerical value is
    divided into two parts. The leading digit(s)
    becomes the stem and the trailing digit the leaf.
    The stems are located along the vertical axis,
    and the leaf values are stacked against each
    other along the horizontal axis.
  • Two disadvantages to organizing the data into a
    frequency distribution
  • The exact identity of each value is lost
  • Difficult to tell how the values within each
    class are distributed.

EXAMPLE Listed in Table 41 is the number of
30-second radio advertising spots purchased by
each of the 45 members of the Greater Buffalo
Automobile Dealers Association last year.
Organize the data into a stem-and-leaf display.
Around what values do the number of advertising
spots tend to cluster? What is the fewest number
of spots purchased by a dealer? The largest
number purchased?
Boxplot - Example
Step1 Create an appropriate scale along the
horizontal axis. Step 2 Draw a box that starts
at Q1 (15 minutes) and ends at Q3 (22 minutes).
Inside the box we place a vertical line to
represent the median (18 minutes). Step 3
Extend horizontal lines from the box out to the
minimum value (13 minutes) and the maximum value
(30 minutes).
Box Plot Buffalo Automobile Example SPSS
Scatter Plots
  • Scatter Plots are used to identify and report any
    underlying relationships among pairs of data
  • The plot consists of a scatter of points, each
    point representing an observation.

Describing Relationship between Two Variables
Scatter Diagram Examples
Describing Relationship between Two Variables
Scatter Diagram Example
  • In the data from AutoUSA presented in the file
    whitner.sav, the information concerned the prices
    of 80 vehicles sold last month at the Whitner
    Autoplex lot in Raytown, Missouri. The data shown
    include the selling price of the vehicle as well
    as the age of the purchaser.
  • Is there a relationship between the selling price
    of a vehicle and the age of the purchaser?
  • Would it be reasonable to conclude that the more
    expensive vehicles are purchased by older buyers?

Describing Relationship between Two Variables
Scatter Diagram SPSS Example