introduction and descriptive statistics. statistics is a science that helps us make better decisions...
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Introduction and Introduction and Descriptive StatisticsDescriptive Statistics
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.
WHAT IS STATISTICSWHAT IS STATISTICS??
STATISTICS The science of collecting, organizing,
presenting, analyzing, and interpreting data to assist in
making more effective decisions.
STATISTICS The science of collecting, organizing,
presenting, analyzing, and interpreting data to assist in
making more effective decisions.
Using Statistics (Two Categories)Using Statistics (Two Categories)
Inferential Statistics Predict and forecast
value of population parameters
Test hypothesis about value of population parameter based on
sample statistic Make decisions
Descriptive Statistics Collect Organize Summarize Display Analyze
A population consists of the set of all measurements for which the investigator is interested.
A sample is a subset of the measurements selected from the population.
A census is a complete enumeration of every item in a population.
Samples and PopulationsSamples and Populations
Census of a population may be: Impossible Impractical Too costly
Why Sample?Why Sample?
Parameter Versus StatisticParameter Versus Statistic
PARAMETER A measurable characteristic of a
population.
STATISTIC A measurable characteristic of a sample.
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
Types of Data - Two TypesTypes of Data - Two Types
• 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 value. Salaries
Scales of MeasurementScales of Measurement
Population MeanPopulation MeanFor 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.
EXAMPLE:
The MedianThe Median
PROPERTIES OF THE MEDIAN1. There is a unique median for each data set.2. It is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such
values occur.3. It can be computed for ratio-level, interval-level, and ordinal-level data.4. It can be computed for an open-ended frequency distribution if the median does not lie in an open-ended class.
EXAMPLES:
MEDIAN The midpoint of the values after they have been ordered from
the smallest to the largest, or the largest to the smallest.
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 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
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 ModeThe Mode
MODE The value of the observation that appears most
frequently.
Measures of DispersionMeasures 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 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.
RANGE
MEAN DEVIATION
VARIANCE AND
STANDARD DEVIATION
EXAMPLE – Mean DeviationEXAMPLE – Mean Deviation
EXAMPLE:
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
505
8060504020
n
xx
Variance and Standard DeviationVariance and Standard Deviation
VARIANCE The arithmetic mean of the squared deviations from the mean.
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 small. 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
STANDARD DEVIATION The square root of the variance.
EXAMPLE – Population Variance and Population EXAMPLE – Population Variance and Population Standard DeviationStandard 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 difference.
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.
2912
348
12
1034...1719
N
x
12412
488,1)( 22
N
X
Sample Variance and Sample Variance and Standard DeviationStandard Deviation
sample the in nsobservatio of number the is
sample the of mean the is
sample the in nobservatio each of value the is
variance sample the is
:Where2
n
X
X
s
EXAMPLE
The hourly wages for a sample of part-
time employees at Home Depot are:
$12, $20, $16, $18, and $19.
What is the sample variance?
EXAMPLE:Determine the arithmetic mean vehicle selling
price given in the frequency table below.
The Arithmetic Mean and Standard The Arithmetic Mean and Standard Deviation of Grouped DataDeviation of Grouped Data
EXAMPLE
Compute the standard deviation of the vehicle selling prices in the frequency table below.
Dividing data into groups or classes or intervals Groups should be:
Mutually exclusive Not overlapping - every observation is assigned to only one
group
Exhaustive Every observation is assigned to a group
Equal-width (if possible) First or last group may be open-ended
Group Data and the HistogramGroup Data and the Histogram
Table with two columns listing: Each and every group or class or interval of values Associated frequency of each group
Number of observations assigned to each group Sum of frequencies is number of observations
N for population n for sample
Class midpoint is the middle value of a group or class or interval
Relative frequency is the percentage of total observations in each class Sum of relative frequencies = 1
Frequency DistributionFrequency Distribution
x f(x) f(x)/nSpending Class ($) Frequency (number of customers) Relative Frequency
0 to less than 100 30 0.163100 to less than 200 38 0.207200 to less than 300 50 0.272300 to less than 400 31 0.168400 to less than 500 22 0.120500 to less than 600 13 0.070
184 1.000
x f(x) f(x)/nSpending Class ($) Frequency (number of customers) Relative Frequency
0 to less than 100 30 0.163100 to less than 200 38 0.207200 to less than 300 50 0.272300 to less than 400 31 0.168400 to less than 500 22 0.120500 to less than 600 13 0.070
184 1.000
• Example of relative frequency: 30/184 = 0.163 • Sum of relative frequencies = 1
Example : Frequency DistributionExample : Frequency Distribution
x F(x) F(x)/nSpending Class ($) Cumulative Frequency Cumulative Relative Frequency
0 to less than 100 30 0.163100 to less than 200 68 0.370200 to less than 300 118 0.641300 to less than 400 149 0.810400 to less than 500 171 0.929500 to less than 600 184 1.000
x F(x) F(x)/nSpending Class ($) Cumulative Frequency Cumulative Relative Frequency
0 to less than 100 30 0.163100 to less than 200 68 0.370200 to less than 300 118 0.641300 to less than 400 149 0.810400 to less than 500 171 0.929500 to less than 600 184 1.000
The cumulative frequency of each group is the sum of the frequencies of that and all preceding groups.
The cumulative frequency of each group is the sum of the frequencies of that and all preceding groups.
Cumulative Frequency DistributionCumulative Frequency Distribution
A histogram is a chart made of bars of different heights. 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
HistogramHistogram
Frequency Histogram
Histogram ExampleHistogram Example
Relative Frequency Histogram
Chebyshev’s Theorem and Empirical RuleChebyshev’s Theorem and Empirical Rule
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 percentiles.
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
Quartiles, Deciles and PercentilesQuartiles, Deciles and Percentiles
Percentiles - ExamplePercentiles - ExampleEXAMPLEListed below are the commissions earned last month by a sample of 15 brokers at Salomon Smith Barney’s 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 value
$1,460 $1,471 $1,637 $1,721 $1,758 $1,787 $1,940 $2,038$2,047 $2,054 $2,097 $2,205 $2,287 $2,311 $2,406
Step 2: Compute the first and third quartiles. Locate L25 and L75 using:
205,2$
721,1$
12100
75)115(4
100
25)115(
75
25
7525
L
L
LL
lyrespective positions,
12th and 4th the at located are quartiles third and first the Therefore,
Range Difference between maximum and minimum values
Interquartile Range Difference between third and first quartile (Q3 - Q1)
Variance Average*of the squared deviations from the mean
Standard Deviation Square root of the variance
Definitions of population variance and sample variance differ slightly.
Measures of Variability or DispersionMeasures of Variability or Dispersion
SkewnessSkewness
Another characteristic of a set of data is the shape. There are four shapes commonly observed: symmetric, positively skewed, negatively skewed, bimodal.
The coefficient of skewness can range from -3 up to 3. A value near -3, indicates considerable negative skewness. A value such as 1.63 indicates moderate positive skewness. 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 present.
The Relative Positions of the Mean, The Relative Positions of the Mean, Median and the ModeMedian and the Mode
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 histogram
• Box Plots Median Lower and upper quartiles Maximum and minimum
Methods of Displaying DataMethods of Displaying Data
Pie ChartPie Chart
33.0%
23.0%
19.0%
19.0%
6.0%
Category
Happy with career
Don't like my job but it is on my career pathJob is OK, but it is not on my career path
Enjoy job, but it is not on my career pathMy job just pays the bills
Figure 1-10: Twentysomethings split on job satisfication
My job just pays the bills
Happy with career
Enjoy job, but it is not on my career path
Job OK, but it is not on my career path
Do not like my job, but it is on my career path
33.0%
23.0%
19.0%
19.0%
6.0%
Category
Happy with career
Don't like my job but it is on my career pathJob is OK, but it is not on my career path
Enjoy job, but it is not on my career pathMy job just pays the bills
Figure 1-10: Twentysomethings split on job satisfication
My job just pays the bills
Happy with career
Enjoy job, but it is not on my career path
Job OK, but it is not on my career path
Do not like my job, but it is on my career path
Bar ChartBar Chart
C41Q4Q3Q2Q1Q
1.5
1.2
0.9
0.6
0.3
0.0
Figure 1-11: SHIFTING GEARS
2003 2004
Quartely net income for General Motors (in billions)
C41Q4Q3Q2Q1Q
1.5
1.2
0.9
0.6
0.3
0.0
Figure 1-11: SHIFTING GEARS
2003 2004
Quartely net income for General Motors (in billions)
Relative Frequency Polygon Ogive
Frequency Polygon and OgiveFrequency Polygon and Ogive
50403020100
0.3
0.2
0.1
0.0
Re
lativ
e F
req
ue
ncy
Sales50403020100
1.0
0.5
0.0
Cu
mu
lativ
e R
ela
tive
Fre
qu
en
cy
Sales
(Cumulative frequency or relative frequency graph)
OSAJJMAMFJDNOSAJJMAMFJDNOSAJJMAMFJ
8 . 5
7 . 5
6 . 5
5 . 5
M o n t h
Mill
ion
s o
f T
on
s
M o n t h l y S t e e l P r o d u c t i o n
Time Series PlotTime Series Plot
Stem-and-Leaf DisplayStem-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: (1) The exact identity of each value is lost (2) Difficult to tell how the values within each class are distributed.
EXAMPLE
Listed in Table 4–1 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 - ExampleBoxplot - 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 – Box Plot – Buffalo Automobile Example – SPSS outputSPSS output
45N =
NUMBERS
180
160
140
120
100
80
Scatter Plots Scatter Plots
• Scatter Plots are used to identify and report any underlying relationships among pairs of data sets.
• The plot consists of a scatter of points, each point representing an observation.
Describing Relationship between Two Describing Relationship between Two Variables – Scatter Diagram ExamplesVariables – Scatter Diagram Examples
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 – Describing Relationship between Two Variables – Scatter Diagram ExampleScatter Diagram Example
Vehicle Age Vs. Selling Price
AGE
6050403020
Pri
ce
($0
00
)
40
30
20
10
Describing Relationship between Two Variables – Scatter Diagram SPSS
Example