Introduction and Descriptive Statistics. Statistics is a science that helps us make better decisions...

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


19, 20, 21, 22, 25.

Thus the median is 21.

The heights of four basketball players, in inches, are:

76, 73, 80, 75


73, 75, 76, 80.

Thus the median is 75.5

The heights of four basketball players, in inches, are:

76, 73, 80, 75


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


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


x F(x) F(x)/nSpending Class ($) Cumulative Frequency Cumulative Relative Frequency


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

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