STAT 211 – 019 Dan Piett West Virginia University Lecture 1.
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Transcript of STAT 211 – 019 Dan Piett West Virginia University Lecture 1.
STAT 211 – 019Dan Piett
West Virginia University
Lecture 1
Overview1.1 Science of Statistics1.2 Displaying Small Sets of Numbers1.3 Graphing Categorical DataBREAK - Problems1.4 Frequency Histograms1.5 Density Histograms1.6 Centers of Data1.7 Mean vs Median
Section 1.1
Science of Statistics
DefinitionsStatistics
The science of collecting, organizing, and analyzing data for the purpose of estimation and making inferences.
DataValues which arise from observing characteristics on
a selected group (sample) of individuals. The characteristic(s) which are observed are called variables.
PopulationThe entire group of “things” from which data may be
collected.Sample
The selected group of the population that is studied.
Population vs. SamplePopulation
All undergraduate students at WVU
SampleA group of 100
undergraduate students at WVU
All WVU Students (population)
Group of 100
Students
(sample)
Types of DataVariables
The characteristics which we look at when observing data. These can be classified into four groups.
NumericDiscreteContinuous
CategoricalRanked (Ordinal)Unranked (Nominal)
Numeric VariablesNumeric Variables
Variables whose values represent quantities. This can be further broken down into discrete and continuous.
Discrete NumericVariables which usually arise by countingExamples
Number of cars in a parking lotNumber of courses taken in a semester
Continuous NumericVariables which arise by measuringExamples
Time spent running a marathonWeight
Categorical VariablesCategorical Variables
Variables that are not numeric. These can be further classified into ranked and unranked.
Ranked Categorical VariablesIf the possible values of a categorical value
follow a “natural” orderingExample
Olympic MedalUnranked Categorical Variables
Anything that is not rankedExample
Blood Type
Two Major Branches of Statistics
Descriptive Statistics Inferential Statistics
Use graphical displays and numeric summarizations to represent data.
First half of this semester
Use analytic methods and theory of probability to draw conclusions or make decisions.
Second half of this semester
Section 1.2
Displaying Small Sets of Numbers
Displaying Small Sets of DataWe will be looking at three different ways
to display small sets of data.Dot PlotsStem and Leaf DisplayHistograms
Dot PlotEach Dot
represents one data value
Example – Height of Students62, 71, 65, 68, 64, 72, 66, 68, 70, 67, 67, 68, 64, 65, 68
Stem and Leaf DisplayRepresents the data using the actual digits
that make up the data.Leading digit becomes the stemTrailing digit becomes the leaf
Stem and Leaf ExamplesExample – Exam
Grades76, 74, 82, 96, 66, 76, 93, 86, 84, 62, 82, 75, 58, 71, 73, 79, 65, 80
Example - Decimals1.3, 2.4, 1.7, 3.2, 5.6
OutliersOutlier
An observation whose value is unusual or extreme.
Example15, 21, 13, 18, 23, 19, 26, 16, 71, 22, 14, 21
71 is an outlier
Section 1.3
Graphing Categorical Data
Grouped Frequency TableFrequencies are
tabulated for each value of the categorical variable.
Example – STAT 101 Grade Distribution (2000)
Bar GraphRepresents categorical
data by showing the amount of data that belong to each category as proportionally sized rectangles.
Example: US Coal Production (in millions)
WV – 172.0PA – 189.2KY – 154.8WY – 233.6Other – 120.4
Pie Chart (Circle Graph)A pie chart shows the
amount of data that belongs to each category as a proportional part of a circle.
Example: US Coal Production (in millions)
WV – 172.0PA – 189.2KY – 154.8WY – 233.6Other – 120.4
Section 1.4
Frequency Histograms
Grouped Frequency DistributionsGrouped Frequency Distributions
A list (or table) which pairs ranges for values of a variable with their frequencies (counts). Each range is called a class.
Rules for constructing a grouped frequency distribution1.Each class should be the same width.2.Classes should not overlap.3.Each observation falls into one and only one
class.4.Use between 3 and 15 classes.
Frequency HistogramFrequency Histogram
A graphical representation of a frequency distribution
Example – Math3 Exam Scores
Shape of a DistributionSymmetric Shaped
Mound Shaped
U-Shaped
Uniform
Shape of a DistributionAsymmetric
Shaped
Skewed Right (Positive)
Skewed Left (Negative)
Break
Section 1.5
Density Histograms
Density HistogramRelative Frequency Histogram
A histogram in which the vertical axis represents percentages or proportions, rather than counts.
Example – Math3 Exam Scores
Constructing Density Histograms1. For each class, compute the percent of
observations in each class.2. Divide the percent associated with each
class by the width of that class. This yields the percent of observations associated with each unit of the measurement scale.
3. Draw the histogram using the values computed in step 2.
More on Density HistogramsIf the bars of a histogram are arranged
from largest to smallest, it is called a Pareto Chart.
A density histogram is a histogram whose vertical axis is scaled so that the sum of the areas of it’s rectangles is 1 square unit.
If the sample size, n, is very large, a density histogram can be used to estimate the distribution of the population from which the data was obtained.
Section 1.6
Centers of Data
AveragesAverage
An average is a single value which represents all of the data.
Types of AveragesSample MeanSample MedianSample Mode
Sample Mean (x)Sample MeanFormula
Examplex: 14, 23, 8, 19, 41Sample Mean = (14 + 23 + 8 + 19 +
41)/5= 105/5= 21
More on Sample MeansIf we acquire data from all members of a
population, we can compute the population mean ( )
Because of this, we can use the sample mean to estimate the population mean
Sample Median ( )Sample Median
The middle value of the observed data values ranked from lowest to highest
If the sample contains n observations, the median is the ½(n+1) ranked value
Example:x: 14, 23, 8, 19, 41Ranked: 8, 14, 19, 23, 41 n = 5Position of the median: ½(5+1) observation =
3rd Median = 19
Sample Median Even ExampleIn the previous example, n was oddExample with an even n:x: 43, 26, 37, 19, 52, 80Ranked: 19, 26, 37, 43, 52, 80 n=6Position of the Median: ½ (6+1) observation
=3½ positionMedian is the mean of the 3rd and 4th ranked
observationMedian = (37+43)/2 = 40
Sample ModeSample Mode
The observed value which occurs with the greatest frequency
Example:x: 4, 3, 1, 4, 0, 3, 3, 1, 4, 0, 1, 1, 2, 2Mode = 1
More Mode ExamplesA set of x might
have no mode (every value occurs once)Mode = NoneNOT Mode = 0
A set of x might have more than one modeMode = 5 and 6
Section 1.7
Mean vs. Median
Best Measure of the Center
Mean Median
Use for approximately symmetric distributions
Greatly influenced by outliers
Use for significantly skewed distributions
Not greatly influenced by outliers
How to Handle OutliersIf the outlier is a “mistaken” data value,
eliminate it from the analysis
If the outlier is an actual data value, do not eliminate it
End of Class