Displaying Distributions with Graphs

Displaying Distributions with Graphs - Transcript

1 1 Displaying Distributions with Graphs
Graphs for Categorical Variables
Categorical variable places an individual into group or category
Pie Chart
Must include all categories
Use to emphasize category s relation to whole
Bar Graphs
Easier to make and read
More flexible can compare any set of quantities that are measured in the same units
Graphs for Quantitative Variables
Quantitative variable takes numerical values for which arithmetic makes sense
Stemplots
Gives picture of shape of distribution
Stem
All but the final digit
May have as many digits as needed
Leaf
The final digit
Contains only one single digit
Back to back stemplot
Compare two related distributions
Common stem with leaves on each side
DO NOT work well for large data sets
Split stems
Trimming numbers by removing last digit s
Histograms
Breaks range of values of variable into classes
Displays only the count or percentage that fall into each class
Choose classes of equal width
Area will then be determined by height
All classes will be fairly represented
Use your judgment in choosing classes to display shape
Technology Toolbox pp 59 60
Examining Distributions
Overall patterns and deviations
Shape center and spread
Center midpoint
Spread smallest and largest values
Shape
Modes one unimodal or several peaks
Symmetric larger and smaller values are mirror images around the midpoint
Skewed
Right skewed the right tail is much longer than the left
Left skewed the left tail is much longer than the right
Outliers an individual value that falls outside the overall pattern
Dealing with Outliers
A matter of judgment
Look for points that are clearly apart from the body of data
In general it is not a good idea to just delete or ignore outliers
Relative Frequency and Cumulative Frequency
Relative Frequency
Divide the count in each class interval by the total count
Multiply by 100 to get the percent
Cumulative Frequency
Add the counts in the frequency column that fall in or below the current interval
Divide the entries by the total count
Multiply by 100 to get the percent
Time Plots
Plots each observation against the time at which it was measured
ALWAYS time on the horizontal axis
Measured variable on vertical axis
Connect the data points with lines to emphasize change over time

1 2 Describing Distributions with Numbers
Measuring Center The Mean
Mean average value

Sensitive to the influence of a few extreme observations
Outliers
Skewed distribution
Not a resistant measure of center
Measuring Center The Median
Formal version of midpoint
Median M the number such that half the observations are smaller and the other half are larger
To find the median of a distribution
Arrange all the observations from smallest to largest
observations up from the bottom of the list
observations up from the bottom of the list to get the location of the median
Mean vs Median
Mean and median are most common measures of center
Mean and median are close together in symmetric distributions
In skewed distributions the mean is farther out in the long tail
The most useful numerical description of a distribution gives both a measure of center and spread
Measuring Spread Range
Range Max Min
Shows full spread of data
Dependent on smallest and largest observations may be outliers
Measuring Spread The Quartiles
Arrange observations in increasing order and locate the median
First quartile 25th percentile median of the lower 50 of observations
Third quartile 75th percentile median of the upper 50 of observations
The Five Number Summary and Boxplots
Consists of the smallest observation min the first quartile Q1 the median M the third quartile Q3 and the largest observation max
Visual representation of the five number summary
The 1 5 X IQR Rule for Suspected Outliers
Distance between quartiles IQR is resistant to outliers
Interquartile range IQR Q3 Q1
IQR alone is not very useful for describing skewed distributions
Suspected outliers
Q3 1 5 X IQR
Q1 1 5 X IQR
Technology Toolbox pp 81 82
Measuring Spread The Standard Deviation
Variance the average of the squares of the deviations from the mean
How far the observations are from their mean
n observations x1 x2 xn

Standard deviation
the square root of the variance

Properties of the Standard Deviation
s measures spread about the mean use only when mean is used as measure of center
s 0 when there is no spread all observations have the same value
s is not resistant
a few outliers can make s very large
s2 makes this measure even more sensitive to a few extreme observations
Choosing Measures of Center and Spread
For skewed distributions or distributions with strong outliers use the five number summary
For reasonably symmetric distributions that are free of outliers use mean and standard deviation
Always plot your data
Graphs give best overall picture of a distribution
Numerical measures of center and spread only give specific facts do not describe its entire shape
Changing the Unit of Measurement
Linear Transformations

Adding a shifts all values of x up or down
Multiplying by b changes the size of the unit of measure
Comparing Distributions
Data
Who
What
Why
When where how and by whom
Graphs
Numerical Summaries
Interpretation Answer the question
Chapter 1 Exploring Data
Mrs S Smith September 2008 Page PAGE MERGEFORMAT 1