Chapter 1 Getting Started Understanding Basic Statistics Fifth Edition By Brase and Brase Prepared...

Chapter 1

Getting Started

Understanding Basic Statistics Fifth Edition

By Brase and Brase Prepared by Jon Booze

© Cengage Learning. All rights reserved. 1 | 2

What is Statistics?

• Collecting data• Organizing data• Analyzing data• Interpreting data


Individuals and Variables

• Individuals are people or objects included in the study.

• Variables are characteristics of the individual to be measured or observed.


Variables

• Quantitative Variable – The variable is numerical, so operations such as adding and averaging make sense.

• Qualitative Variable – The variable describes an individual through grouping or categorization.


Data

• Population Data – The data are from every individual of interest.

• Sample Data – The data are from only some of the individuals of interest.


DataWhich of the following Venn diagrams shows the

relationship between population data and sample data?

a). b).

c). d).

S P

S

P S

P

P

S


Levels of Measurement

• Nominal Level – The data consists of names, labels, or categories.

• Ordinal Level – The data can be ordered, but the differences between data values are meaningless.



• Interval Level – The data can be ordered and the differences between data values are meaningful.

• Ratio Level – The data can be ordered, differences and ratios are meaningful, and there is a meaningful zero value.



The freezing points of four liquids are 32°F, 6°F, 13°F, and 20°F. What is the level of these measurements?

a). Nominalb). Ordinalc). Intervald). Ratio


Two Branches of Statistics

• Descriptive Statistics: Organizing, summarizing, and graphing information from populations or samples.

• Inferential Statistics: Using information from a sample to draw conclusions about a population.


Sampling Techniques• Simple Random Sampling, Sample size = n

– Each member of the population has an equal chance of being selected.

– Each sample of size n has an equal chance of being selected.

• Stratified sampling Population

Subgroup 4

Subgroup 1Subgroup 2Subgroup 3

sample


Sampling Techniques • Systematic sampling

– Number every member of the population.– Select every kth member.

• Cluster sampling– Population is naturally divided into pre-

existing segments.– Make a random selection of clusters, then

select all members of each cluster.

• Convenience sampling - Collect sample data from a readily-available population database.


Critical Thinking

Which of the following sampling strategies is likely to lead to a non-sampling error?

Individuals are selected at random from…a). A database of social security numbers.b). A cluster of phone books.c). A collection of birth certificates.d). None of these is likely to introduce non-

sampling error.


Critical Thinking

Which of the following sampling strategies is likely to lead to a non-sampling error?

Individuals are selected at random from…a). A database of social security numbers.b). A cluster of phone books.c). A collection of birth certificates.d). None of these is likely to introduce non-sampling

error.Not everyone has a phone. Sampling from phone

books may introduce bias.


Guidelines For Planning a Statistical Study

1. Identify individuals or objects of interest.2. Specify the variables.3. Determine if you will use the entire

population. If not, determine an appropriate sampling method

4. Determine a data collection plan, addressing privacy, ethics, and confidentiality if necessary.


Guidelines For Planning a Statistical Study

5. Collect data.6. Analyze the data using appropriate statistical

methods.7. Note any concerns about the data and

recommend any remedies for further studies.


Census vs. Sample

• In a census, measurements or observations are obtained from the entire population (uncommon and often impractical).

• In a sample, measurements or observations are obtained from part of the population (common).


Observational Studies and Experiments

• Observational Study – Measurements are obtained in a way that does not change the response or the variable being measured. (No treatment is applied.)

• Experiment – A treatment is applied in order to observe its effect on the variable being measured.


Experiment

• Used to determine the effect of a treatment.

• Experimental design needs to control for other possible causes of the effect.

– Placebo effect. – Lurking variables.

• To minimize these confounds, create one or more control groups that receive no treatment.


Experiment Designs

• Double-Blinding – minimizes the unintentional transfer of bias between researcher and subject.


Surveys• Collecting data from respondents by asking them

questions.

Survey Pitfalls• Nonresponse → undercoverage of population.• Truthfulness – respondents sometimes lie.• Faulty recall of respondent• Hidden bias – due to poor question wording.• Vague wording – “sometimes”, “often”, “seldom”• Interviewer influence – who is asking the

questions and in what manner.• Voluntary response – relatively interested

individuals are more likely to participate.

Chapter 2

Organizing Data




Frequency Tables

• A frequency table– organizes quantitative data.– partitions data into classes (intervals).– shows how many data values are in each

class.

Test Score Number of Students

61-70 4

71-80 8

81-90 15

91-100 7


Data Classes and Class Frequency• Class: an interval of values.

– Example: 61 x 70

• Frequency: the number of data values that fall within a class.

– “Five data fall within the class 61 x 70”.

• Relative Frequency: the proportion of data values that fall within a class.

– “18% of the data fall within the class 61 x 70”.


Structure of a Data ClassA “data class” is basically an interval on a number

line.

It has:• A lower limit a and an upper limit b.• A width.• A lower boundary and an upper boundary (integer data).• A midpoint.


Structure of a Data ClassA “data class” is basically an interval on a number

line.

If a = 60 and b = 69 for integer data, what is the value of the lower boundary?

a). 60 b). 59.5

c). 9 d). 64.5


Constructing Data Classes• Find the class width.

–

– Increase the computed value to the next higher whole number.

• Find the class limits. – The lower limit of the “leftmost” class is set

equal to the smallest value in the data set.

Largest data value – smallest data valueDesired number of classes


Constructing Data Classes, cont’d

• Find the class boundaries (integer data).– Subtract 0.5 from the lower class limit and

add 0.5 to the upper class limit.

For a certain data set, the minimum value is 25 and the maximum value is 58. If you wish to partition the data into 5 classes, what would be the class width?

a). 5 b). 6 c). 7 d). 8


Histograms• Histogram – graphical summary of a frequency

table.• Uses bars to plot the data classes versus the

class frequencies.


Making a Histogram• Make a frequency table.

• Place class boundaries on horizontal axis. Place frequencies on vertical axis.

• For each class, draw a bar with height equal to the class frequency and width equal to the class width plus 1.


Making a Histogram


Distribution Shapes

Symmetric Uniform

Skewed Left Skewed Right

Bimodal


Graphical Displays…

• … represent the data.

• … induce the viewer to think about the substance of the graphic.

• …should avoid distorting the message of the data.


Bar Graphs

• Used for qualitative or quantitative data.

• Can be vertical or horizontal.

• Bars are uniformly spaced and have equal widths.

• Length/height of bars indicate counts or percentages of the variable.

• “Good practice” requires including titles and units and labeling axes.


Bar Graphs

Example:


Pareto Charts

• A bar chart with two specific features:

– Heights of bars represent frequencies.

– Bars are vertical and are ordered from tallest to shortest.


Circle Graphs/Pie Charts• Used for qualitative data

• Wedges of the circle represent proportions of the data that share a common characteristic.

• “Good practice” requires including a title and either wedge labels or legend.


Time-Series• Shows data measurements in chronological

order.

• Data are plotted in order of occurrence at regular intervals over a period of time.


Critical Thinking – which type of graph to use?

• Bar graphs are useful for quantitative or qualitative data.

• Pareto charts identify the frequency in decreasing order.

• Circle graphs display how a total is dispersed into several categories.

• Time-series graphs display how data change over time.



What type of graph would be best for showing the ice cream flavor preferences of a group of 100 children?

a). Histogram b). Pareto graphc). Time series graph d). Circle graph


Stem and Leaf Plots• Displays the distribution of the data while

maintaining the actual data values.

• Each data value is split into a stem and a leaf.


Stem and Leaf Plot Construction


Critical Thinking

• Large gaps between stems containing leaves, especially at the top or bottom, suggest the existence of outliers.

• Watch the outliers – are they data errors or simply unusual data values?

Chapter 3

Averages and Variation




Measures of Central Tendency

• Average – a measure of the center value or central tendency of a distribution of values.

• Three types of average:– Mode– Median– Mean


ModeThe mode is the most frequently occurring value in

a data set.

Example: Sixteen students are asked how many college math classes they have completed.

{0, 3, 2, 2, 1, 1, 0, 5, 1, 1, 0, 2, 2,

7, 1, 3}

The mode is 1.


Median

Finding the median:1). Order the data from smallest to largest.

2). For an odd number of data values:Median = Middle data value

3). For an even number of data values:Sum of middle two valuesMedian

2


Sample mean Population mean

Mean

xx

n x

N


Resistant Measures of Central Tendency

• A resistant measure will not be affected by extreme values in the data set.

• The mean is not resistant to extreme values.

• The median is resistant to extreme values.

• A trimmed mean is also resistant.


Critical Thinking

• Four levels of data – nominal, ordinal, interval, ratio (Chapter 1)

• Mode – can be used with all four levels.

• Median – may be used with ordinal, interval, of ratio level.

• Mean – may be used with interval or ratio level.


Critical Thinking

• Mound-shaped data – values of mean, median and mode are nearly equal.


Measures of Variation

• Range = Largest value – smallest value

Only two data values are used in the computation, so much of the information in the data is lost.

Three measures of variation: rangevariancestandard deviation


The Coefficient of Variation

100x

sCV 100

CV

For Samples For Populations


Percentiles and Quartiles

• For whole numbers P, 1 ≤ P ≤ 99, the Pth percentile of a distribution is a value such that P% of the data fall below it, and (100-P)% of the data fall at or above it.

• Q1 = 25th Percentile• Q2 = 50th Percentile = The Median• Q3 = 75th Percentile


Quartiles and Interquartile Range (IQR)


Computing Quartiles


Five Number Summary

• A listing of the following statistics:

– Minimum, Q1, Median, Q3, Maximum

• Box-and-Whisder plot – represents the five-number summary graphically.


Box-and-Whisker Plot Construction


• Box-and-whisker plots display the spread of data about the median.

• If the median is centered and the whiskers are about the same length, then the data distribution is symmetric around the median.

• Fences – may be placed on either side of the box. Values lie beyond the fences are outliers. (See problem 10)

Critical Thinking


Problems

• Pg. 109 #4, 5

Chapter 1 Getting Started Understanding Basic Statistics Fifth Edition By Brase and Brase Prepared...

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