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Chapter 1

Introduction to Statistics

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Datacollections 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

SampleThat part of the population from which information is

obtained.

PopulationThe 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 1.1Relationship 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 whicheach possible sample of a given size is equally likely to bethe one obtained.

Simple random sample: A sample obtained by simplerandom 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

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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.

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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

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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)

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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,…)

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Interval LevelInterval 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 LevelRatio 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 Data1. 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.

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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 LimitsThe Lower class limits are the smallest numbers that can actually

belong to different classes.

Lower ClassLimits

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

belong to different classes.

Upper ClassLimits

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Class BoundariesThe 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

ClassBoundaries

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Class Midpoints

64.5

74.5

84.5

94.5

104.5

114.5

124.5

ClassMidpoints

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Class WidthClass 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

percentagefrequency

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

cies

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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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HistogramBasically a graphic version of a frequency distribution.

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HistogramThe 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.

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Histogram

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Frequency PolygonUses line segments connected to points directly above class midpoint values.

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Relative Frequency PolygonUses 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 ChartA bar graph for qualitative data, with the bars arranged in descending order according to frequencies

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Pie ChartA 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.