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Chapter 1

Introduction to Statistics

Slide 1-2

Data

collections of observations

(such as measurements,

genders, survey responses)

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Statistics

It is the study of the • collection, • organization, • analysis, • interpretation and • presentation of data

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Population, Sample and Census

Sample

That part of the population from which information is

obtained.

Population

The collection of all individuals or items

under consideration in a statistical study.

Census

Collection of data from every member of a population.

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Figure 1.1 Relationship between population and sample

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Parameter

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Statistic

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Simple Random Sampling; Simple Random Sample

There are two types of simple random sampling.

One is simple random sampling with replacement,

whereby a member of the population can be

selected more than once; the other is simple random

sampling without replacement, whereby a member

of the population can be selected at most once.

Simple random sampling: A sampling procedure for which

each possible sample of a given size is equally likely to be

the one obtained.

Simple random sample: A sample obtained by simple

random sampling.

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Basic Data Types

Quantitative ( or numerical or measurement ) data

Categorical (or qualitative or attribute) data

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Quantitative Data

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Categorical Data

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Working with Quantitative Data

Quantitative data can further be described by

distinguishing between discrete and continuous

types.

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Discrete Data

Discrete data result when the number of possible

values is either a finite number or a ‘countable’

number (i.e. the number of possible values is

0, 1, 2, 3, . . .)

Example: The number of eggs that a hen lays, Test

score, shoe size, age, world ranking, number of

brothers etc.

The number of eggs that a hen lays is discrete

quantitative measure because it is numeric but can

only be a whole number

Slide 1-14

Continuous Data

Continuous (numerical) data

result from infinitely many possible values that

correspond to some continuous scale that covers a

range of values without gaps, interruptions, or jumps

Example: Height, weight, length, amounts of milk from cows

etc.

Height is continuous quantitative measure because it can take

any numerical value in a particular range.

The amount of milk that a cow produces; e.g. 2.343115 gallons

per day.

Slide 1-15

Decide whether the following data are qualitative,

discrete quantitative or continuous quantitative.

1. Number of cars

2. Mass of an object

3. distance of FAU from home

4. Day of the week

5. Color of cars

6. Pocket money

7. Favorite soccer team

8. World ranking

9. Birth place

10. Age

Slide 1-16

Classification of Data using levels of measurement

1. Nominal level of measurement

2. Ordinal level of measurement

3. Interval level of measurement

4. Ratio level of measurement

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Nominal Level

Nominal level of measurement is characterized by data

that consist of names, labels, or categories only, and the

data cannot be arranged in an ordering scheme (such as

low to high)

Examples:

Survey responses yes, no, undecided

Political Party: The political party affiliation of survey

respondents (Democrat, Republican, Independent, other)

Slide 1-18

Ordinal Level

Ordinal level of measurement

involves data that can be arranged in some order, but

differences (obtained by subtraction) between data values

either cannot be determined or are meaningless

Example:

Course grades A, B, C, D, or F

Universities rank in USA (like 1st, 2nd, 3rd, 4th,…)

Slide 1-19

Interval Level

Interval level of measurement is like the ordinal level, with the

additional property that the difference between any two data values is

meaningful. However, data at this level do not have a natural zero

starting point (where none of the quantity is present).

Example:

Body temperatures of 96.2 F and 98.6 F (There is no natural starting

point. The value of 0 F might seem like a starting point, but it is

arbitrary and does not represent the total absence of heat.)

Years: 1000, 2000, 1776, and 1492. (Time did not begin in the year

0, so the year 0 is arbitrary instead of being a natural zero starting

point representing “no time.”)

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Ratio Level Ratio level of measurement Is the interval level with the additional

property that there is also a natural zero starting point (where zero

indicates that none of the quantity is present); for values at this level,

differences and ratios are meaningful.

Example:

Prices: Prices of college textbooks ($0 represents no cost, a $100 book

costs twice as much as a $50 book.)

Distances: Distances (in miles) travelled by cars (0 mile represents no

distance travelled, and 60 miles is twice as far as 30 miles)

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Summary - Levels of Measurement

Nominal - categories only

Ordinal - categories with some order

Interval - differences but no natural

starting point

Ratio - differences and a natural starting

point

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Chapter 2

Summarizing and Graphing Data

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Important Characteristics of Data 1. Center: A representative or average value that

indicates where the middle of the data set is located.

2. Variation: A measure of the amount that the data values vary.

3. Distribution: The nature or shape of the spread of data over the range of values (such as bell-shaped, uniform, or skewed).

0

10

20

30

40

50

60

70

80

90

1st Qtr 2nd Qtr 3rd Qtr 4th Qtr

East

West

North

4. Outliers: Sample values that lie very far away from the vast majority of other sample values.

5. Time: Changing characteristics of the data over time.

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Frequency Distribution (or Frequency Table)

In statistics, a frequency distribution is an

arrangement of the values that one or more

variables take in a sample. Each entry in the

table contains the frequency or count of the

occurrences of values within a particular group

or interval, and in this way, the table

summarizes the distribution of values in the

sample.

Slide 1-25

Pulse Rates of Females and Males

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Frequency Distribution Pulse Rates of Females

The frequency for a particular class is the number of original values that fall into that class.

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Lower Class Limits The Lower class limits are the smallest numbers that can actually

belong to different classes.

Lower Class Limits

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Upper Class Limits The upper class limits are the largest numbers that can actually

belong to different classes.

Upper Class Limits

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Class Boundaries The class boundaries are the numbers used to separate classes, but

without the gaps created by class limits.

59.5

69.5

79.5

89.5

99.5

109.5

119.5

129.5

Class Boundaries

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Class Midpoints

64.5

74.5

84.5

94.5

104.5

114.5

124.5

Class Midpoints

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Class Width Class width is the difference between two consecutive lower

class limits or two consecutive lower class boundaries.

Class Width

10

10

10

10

10

10

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Constructing A Frequency Distribution

3. Starting point: Choose the minimum data value or a convenient

value below it as the first lower class limit.

4. Using the first lower class limit and class width, proceed to list the

other lower class limits.

5. List the lower class limits in a vertical column and proceed to enter

the upper class limits.

6. Take each individual data value and put a tally mark in the appropriate

class. Add the tally marks to get the frequency.

class width (maximum value) – (minimum value)

number of classes

1. Determine the number of classes (should be between 5 and 20).

2. Calculate the class width (round up).

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Relative Frequency Distribution

. includes the same class limits as a frequency

distribution, but the frequency of a class is replaced with

a relative frequencies (a proportion) or a percentage

frequency ( a percent)

relative frequency = class frequency

sum of all frequencies

percentage frequency

class frequency

sum of all frequencies 100% =

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Relative Frequency Distribution

Total Frequency = 40 * 12/40 100 = 30%

*

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Cumulative Frequency Distribution

Cu

mu

lati

ve F

req

uen

cie

s

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Frequency Tables

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Characteristic of Normal Distribution

It has a “bell” shape.

The frequencies start low, then increase to one or two high

frequencies, then decrease to a low frequency.

The distribution is approximately symmetric, with

frequencies preceding the maximum being roughly a mirror

image of those that follow the maximum.

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Histogram

A graph consisting of bars of equal width

drawn adjacent to each other (without gaps).

The horizontal scale represents the classes

of quantitative data values and the vertical

scale represents the frequencies. The

heights of the bars correspond to the

frequency values.

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Histogram Basically a graphic version of a frequency

distribution.

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Histogram

The bars on the horizontal scale are labeled with one of

the following:

(1) Class boundaries

(2) Class midpoints

(3) Lower class limits (introduces a small error)

Horizontal Scale for Histogram: Use class

boundaries or class midpoints.

Vertical Scale for Histogram: Use the class

frequencies.

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Relative Frequency Histogram

It has the same shape and horizontal scale as a histogram, but the

vertical scale is marked with relative frequencies instead of actual

frequencies.

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Interpreting Histograms

When graphed, a normal distribution has a “bell” shape.

Characteristic of the bell shape are

(1) The frequencies increase to a maximum, and then

decrease, and

(2) symmetry, with the left half of the graph roughly a

mirror image of the right half.

The histogram on the next slide illustrates this.

Slide 1-43

Histogram

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Frequency Polygon

Uses line segments connected to points directly above class

midpoint values.

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Relative Frequency Polygon Uses relative frequencies (proportions or percentages) for

the vertical scale.

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Ogive

A line graph that depicts cumulative frequencies

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Dot Plot

Consists of a graph in which each data value is plotted

as a point (or dot) along a scale of values. Dots

representing equal values are stacked.

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Bar Graph

Uses bars of equal width to show

frequencies of categories of qualitative data.

Vertical scale represents frequencies or

relative frequencies. Horizontal scale

identifies the different categories of

qualitative data.

A multiple bar graph has two or more sets of

bars, and is used to compare two or more

data sets.

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Multiple Bar Graph

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Pareto Chart A bar graph for qualitative data, with the bars

arranged in descending order according to

frequencies

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Pie Chart A graph depicting qualitative data as slices of a

circle, size of slice is proportional to frequency count

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Scatter Plot (or Scatter Diagram)

A plot of paired (x,y) data with a horizontal x-axis and a

vertical y-axis. Used to determine whether there is a

relationship between the two variables.

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Time-Series Graph

Data that have been collected at different points in time:

time-series data.