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- Introduction and Descriptive Statistics

WHAT IS 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

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.

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

value. - 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. - EXAMPLE

The Median

- PROPERTIES OF 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. - EXAMPLES

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

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 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

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

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

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

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.

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

below.

- EXAMPLE
- 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

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

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

groups.

Histogram

- 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

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

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

Percentiles - Example

- 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

value - 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.

Skewness

- 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, 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

histogram - Box Plots
- Median
- Lower and upper quartiles
- Maximum and minimum

Pie Chart

Bar Chart

Frequency Polygon and Ogive

(Cumulative frequency or relative frequency

graph)

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

output

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 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