Basic Concepts of Statistics & Its Analysis

©The McGraw-Hill Companies, Inc. 2008
McGraw-Hill/Irwin
Basic Concepts on Statistics
Session 1

2
GOALS
 Understand why we study statistics.
 Explain what is meant by descriptive
statistics and inferential statistics.
 Distinguish between a qualitative variable
and a quantitative variable.
 Describe how a discrete variable is different
from a continuous variable.
 Distinguish among the nominal, ordinal,
interval, and ratio levels of measurement.

3
What is Meant by Statistics?
Statistics is the science of
collecting, organizing, presenting,
analyzing, and interpreting
numerical data to assist in
making more effective decisions.

4
Why Study Statistics?
What can you expect to get out of the study of
business statistics?
 Learn how to present and summarise data in
a meaningful way.
 Learn how to use data to make informed
decisions.
Develop critical and analytical thinking about
data and decisions made from it

5
Who Uses Statistics?
Statistical techniques are used
extensively by marketing,
accounting, quality control,
consumers, professional sports
people, hospital administrators,
educators, politicians, physicians,
etc...

6
Types of Statistics – Descriptive
Statistics
Descriptive Statistics - methods of organizing,
summarizing, and presenting data in an
informative way.
EXAMPLE 1: A Gallup poll found that 49% of the people in a survey knew the name of
the first book of the Bible. The statistic 49 describes the number out of every 100
persons who knew the answer.
EXAMPLE 2: According to Consumer Reports, General Electric washing machine
owners reported 9 problems per 100 machines during 2001. The statistic 9
describes the number of problems out of every 100 machines.
Inferential Statistics: A decision, estimate,
prediction, or generalization about a
population, based on a sample.

7
Population versus Sample
A population is a collection of all possible individuals, objects, or
measurements of interest.
A sample is a portion, or part, of the population of interest

Parameter and Statistic
 Parameter: A numerical measure that
describes a characteristic of a population.
 Statistic: A numerical measure that describes
a characteristic of a sample
9

11
Types of Variables
A. Qualitative or Attribute variable - the
characteristic being studied is nonnumeric.
EXAMPLES: Gender, religious affiliation, type of automobile
owned, state of birth, eye color are examples.
B. Quantitative variable - information is reported
numerically.
EXAMPLES: balance in your checking account, minutes
remaining in class, or number of children in a family.

12
Quantitative Variables - Classifications
Quantitative variables can be classified as either discrete
or continuous.
A. Discrete variables: can only assume certain values
and there are usually “gaps” between values.
EXAMPLE: the number of bedrooms in a house, or the number of hammers sold at the local
Home Depot (1,2,3,…,etc).
B. Continuous variable can assume any value within a
specified range.
EXAMPLE: The pressure in a tire, the weight of a pork chop, or the height of students in a
class.

13
Summary of Types of Variables

14
Four Levels of Measurement
Nominal level - data that is
classified into categories and
cannot be arranged in any
particular order.
EXAMPLES: eye color, gender,
religious affiliation.
Ordinal level – involves data
arranged in some order, but the
differences between data
values cannot be determined or
are meaningless.
EXAMPLE: During a taste test of
4 soft drinks, Mellow Yellow
was ranked number 1, Sprite
number 2, Seven-up number
3, and Orange Crush number
4.
Interval level - similar to the ordinal
level, with the additional
property that meaningful
amounts of differences between
data values can be determined.
There is no natural zero point.
EXAMPLE: Temperature on the
Fahrenheit scale.
Ratio level - the interval level with
an inherent zero starting point.
Differences and ratios are
meaningful for this level of
measurement.
EXAMPLES: Monthly income
of surgeons, or distance
traveled by manufacturer’s
representatives per month.

15
Summary of the Characteristics for
Levels of Measurement

©The McGraw-Hill Companies, Inc. 2008
McGraw-Hill/Irwin
Describing Data:
Numerical Measures

17
GOALS
• Calculate the arithmetic mean, weighted mean, median, mode.
• Explain the characteristics, uses, advantages, and disadvantages of each
measure of location.
• Identify the position of the mean, median, and mode for both symmetric and
skewed distributions.
• Compute and interpret the range, mean deviation, variance, and standard
deviation.
• Understand the characteristics, uses, advantages, and disadvantages of
each measure of dispersion.
• Understand Chebyshev’s theorem and the Empirical Rule as they relate to a
set of observations.

18
Characteristics of the Mean
The arithmetic mean is the most widely used
measure of location. It requires the interval
scale. Its major characteristics are:
– All values are used.
– It is unique.
– It is calculated by summing the values and
dividing by the number of values.

19
Population Mean
For ungrouped data, the population mean is the
sum of all the population values divided by the
total number of population values:

20
EXAMPLE – Population Mean

21
Sample Mean
 For ungrouped data, the sample mean is the
sum of all the sample values divided by the
number of sample values:

23
Properties of the Arithmetic Mean
 Every set of interval-level and ratio-level data has a mean.
 All the values are included in computing the mean.
 A set of data has a unique mean.
 The mean is affected by unusually large or small data values.
 The arithmetic mean is the only measure of central tendency
where the sum of the deviations of each value from the mean is
zero.

24
Weighted Mean
 The weighted mean of a set of numbers X1,
X2, ..., Xn, with corresponding weights w1, w2,
...,wn, is computed from the following
formula:

25
EXAMPLE – Weighted Mean
The Carter Construction Company pays its hourly
employees $16.50, $19.00, or $25.00 per hour.
There are 26 hourly employees, 14 of which are paid
at the $16.50 rate, 10 at the $19.00 rate, and 2 at the
$25.00 rate. What is the mean hourly rate paid the
26 employees?

26
The Median
The Median is the midpoint of the values
after they have been ordered from the
smallest to the largest.
– There are as many values above the median as
below it in the data array.
– For an even set of values, the median will be the
arithmetic average of the two middle numbers.

27
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.

28
EXAMPLES - Median
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

29
The Mode
 The mode is the value of the observation
that appears most frequently.

Measure of Shape
 Skewness is the extent of asymmetry in the
distribution. If the distribution is symmetric
then it is not skewed.
31

33
The Relative Positions of the Mean,
Median and the Mode

34
Dispersion
Why Study 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.

35
Measures of Dispersion
 Range
 Variance and Standard
Deviation

36
EXAMPLE – Range
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.
Range = Largest – Smallest value
= 80 – 20 = 60

38
EXAMPLE – Variance and Standard
Deviation
The number of traffic citations issued during the last five months in
Beaufort County, South Carolina, is 38, 26, 13, 41, and 22. What
is the population variance?

39
EXAMPLE – Sample Variance
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?

43
Chebyshev’s Theorem
The arithmetic mean biweekly amount contributed by the Dupree
Paint employees to the company’s profit-sharing plan is $51.54,
and the standard deviation is $7.51. At least what percent of
the contributions lie within plus 3.5 standard deviations and
minus 3.5 standard deviations of the mean?

45
 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 and percentiles.
Quartiles and Percentiles

46
 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.
Percentile Computation

47
Percentiles - Example
Listed below are the commissions earned last month by
a sample of 15 brokers at Salomon Smith Barney’s
Oakland, California, office. Salomon Smith Barney is
an investment company with offices located
throughout the United States.
$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.

48
Percentiles – Example (cont.)
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

49
Percentiles – Example (cont.)
Step 2: Compute the first and third quartiles.
Locate L25 and L75 using:
205
,
2
$
721
,
1
$
ly
respective
array,
in the
n
observatio
12th
and
4th
the
are
quartiles
third
and
first
the
Therefore,
12
100
75
)
1
15
(
4
100
25
)
1
15
(
75
25
75
25








L
L
L
L

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