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2 - 2 Copyright © 2014 Pearson Education, Inc. All rights reserved Learning Objectives  Understand that a distribution of a sample of data displays a variable’s values and the frequencies (or relative frequencies) of those values.  Know how to make graphs of distributions of numerical and categorical variables and how to interpret the graphs in context.  Be able to compare centers and spreads of distributions of samples informally.

2 - 4 Copyright © 2014 Pearson Education, Inc. All rights reserved Visualizing Statistics  Organize the data using the chart that most effectively visually summarizes the data.  The distribution of the data describes the values, frequencies (counts), and “shape” of the data.  Is there a data value or data values that are far from the rest of the data?  Is there symmetry?  Is there a most common value or most common range of values?

2 - 5 Copyright © 2014 Pearson Education, Inc. All rights reserved Dot Plots  A Dot Plot is a chart that contains a dot for each data value.  Benefits  Shows the individual data values  Easy to spot outliers  Describes the distribution visually  Drawbacks  Not as common as bar and pie charts  Not great for data that has too many individual values

2 - 6 Copyright © 2014 Pearson Education, Inc. All rights reserved Dot Plot Example  Clearly shows the outlier just below \$300.  The rest of the data is generally uniformly spread out.

2 - 7 Copyright © 2014 Pearson Education, Inc. All rights reserved Frequency Histograms  A histogram is a type of bar graph.  The horizontal axis is numerical.  The vertical axis represents the frequency of the data.  Groups the data into bins, also called intervals or classes.  Easy to visualize the distribution.

2 - 8 Copyright © 2014 Pearson Education, Inc. All rights reserved Histogram Example  Different bin widths depict the same data differently.  The smaller width shows more detail.  Too small a width shows too much detail and will not clearly display the main features.

2 - 9 Copyright © 2014 Pearson Education, Inc. All rights reserved Relative Frequency Histograms  A Relative Frequency Histogram is a histogram where the vertical axis represents the relative frequencies, or percents, rather than the frequencies.  Compute the relative frequency by dividing the frequency by the sample size.  The relative frequency histogram always has the same shape as the frequency histogram. The scale of the vertical axis is just changed.

2 - 10 Copyright © 2014 Pearson Education, Inc. All rights reserved Relative Frequency Example  Clearly shows that half of all women score on average between 0.7 and 0.8 goals per game.  Shows there are a small number of exceptional players. Women’s Soccer Players, NCAA Division III 2009

2 - 11 Copyright © 2014 Pearson Education, Inc. All rights reserved Frequency vs. Relative Frequency Histograms  Use a frequency histogram when you want to emphasize how many are in each range.  Use a relative frequency histogram when you want to emphasize what proportion or percent of the total each range contains.

2 - 12 Copyright © 2014 Pearson Education, Inc. All rights reserved Stem and Leaf Plots  The Leaf is the last digit  The Stem contains all digits before the last digit  Shows individual data values  Same as a histogram, but bin width a power of 10  Example: The five 0’s show that there were five classes with 40 students. Class Size

2 - 14 Copyright © 2014 Pearson Education, Inc. All rights reserved Three Aspects of a Distribution  Shape  Symmetry  How a many bumps or modes?  Other distinguishing features  Center  What is a typical value?  Spread  Is the data all close together or spread out?

2 - 15 Copyright © 2014 Pearson Education, Inc. All rights reserved Skewness  A distribution is Skewed Right if most of the data values are small and there is a “tail” of larger values to the right.  A distribution is Skewed Left if most of the data values are large and there is a “tail” of smaller values to the left.

2 - 16 Copyright © 2014 Pearson Education, Inc. All rights reserved Symmetric Distributions  A distribution is symmetric if the left hand side is roughly the mirror image of the right hand side. Symmetric Distributions

2 - 17 Copyright © 2014 Pearson Education, Inc. All rights reserved How Many Mounds  A Unimodal distribution has one mound.  A Multimodal distribution has more than two mounds.  A Bimodal distribution has two mounds.

2 - 18 Copyright © 2014 Pearson Education, Inc. All rights reserved Normal Distributions  A Normal distribution has the following properties  Symmetric  Unimodal  Mound or Bell Shaped

2 - 19 Copyright © 2014 Pearson Education, Inc. All rights reserved Outliers  An Outlier is a data value that is either much smaller or much larger than the rest of the data.  Some reasons for outliers  Error in data collection  No error. For example, the owner’s salary could be an outlier if the rest of the employees are all low wage workers

2 - 20 Copyright © 2014 Pearson Education, Inc. All rights reserved Center  What is a typical value?  Center not a typical value for bimodal or skewed.

2 - 22 Copyright © 2014 Pearson Education, Inc. All rights reserved Summary of Describing a Distribution  What is the shape?  Is it Symmetric, Skewed, or Neither?  Unimodal, Bimodal, or Multimodal?  Normal?  Are there outliers?  Where is the center? Is the center a typical value?  Is there low or high variability?

2 - 24 Copyright © 2014 Pearson Education, Inc. All rights reserved Two Types of Charts  A Bar Chart is like a histogram, but the horizontal axis can represent categorical data. A natural order may not occur.  A Pie Chart is a circle cut into slices where the size of each slice is proportional to the frequency of the outcome that it represents.

2 - 25 Copyright © 2014 Pearson Education, Inc. All rights reserved  The frequency table below shows the ranks of a group of army members who live in the barracks. Example: Categorical Data RankPrivateCorporalSergeantMajor Frequency 65224916

2 - 26 Copyright © 2014 Pearson Education, Inc. All rights reserved Bar Chart  A graphical summary for categorical data  Each category is represented by a bar.  The height of each bar is proportional to the frequency for that category.  There can be more than one choice of ordering the categories.

2 - 28 Copyright © 2014 Pearson Education, Inc. All rights reserved Pareto Chart  A Pareto Chart is a bar chart that orders the categories from largest to smallest frequency.

2 - 29 Copyright © 2014 Pearson Education, Inc. All rights reserved Differences Between Bar Charts and Histograms  A histogram displays numerical data. A bar chart can display categorical data.  The bar widths of a histogram are meaningful and must all be the same size. The bar widths for a bar chart are meaningless.  The bars of a histogram must touch each other. For a bar chart, there are gaps between bars.  There is only one choice, ascending by x, for the order of the bars, while there are many choices of order for a bar chart.

2 - 30 Copyright © 2014 Pearson Education, Inc. All rights reserved Pie Charts  Graphical summary for categorical data.  A circle is cut into several slices. The size of each slice is proportional to the frequency of the category that it represents.  Often used to display how much of a share each category has of the whole.  If f is the frequency and n is the sample size, the angle of each slice is

2 - 33 Copyright © 2014 Pearson Education, Inc. All rights reserved Description of Numerical Distributions vs. Categorical Distributions  Numerical Distributions  Shape  Center  Spread  Categorical Distributions  Mode  Variability or Diversity

2 - 35 Copyright © 2014 Pearson Education, Inc. All rights reserved Mode  The Mode is the category that occurs with the highest frequency.  The mode is thought of as the typical outcome.  If there is a close tie between two categories for most frequently occurring, the distribution is called bimodal.  If more than two categories have roughly the tallest bars, the distribution is called multimodal.

2 - 38 Copyright © 2014 Pearson Education, Inc. All rights reserved Variability  If the distribution has a lot of diversity (many observations in many different categories), then variability is high.  If the distribution has only a little diversity (many of the observations fall into the same category), then variability is low.  Caution: Variability is about many different categories, not many frequencies.

2 - 43 Copyright © 2014 Pearson Education, Inc. All rights reserved Ways to Mislead with Graphs: Don’t Do Any of These!  Have the frequency scale not begin at 0 to create the illusion of greater differences.  Use symbols other than bars that hide or accentuate the real differences.  Use unequal width bars.

2 - 46 Copyright © 2014 Pearson Education, Inc. All rights reserved Scale Unclear The scale is by area and not by just height. 20052006 20072008 Homes Sold by Year

2 - 47 Copyright © 2014 Pearson Education, Inc. All rights reserved Other Creative Charting Techniques  Internet and computers allow for additional effects  Analysis of State of the Union Speeches Analysis of State of the Union Speeches  World Population Changes World Population Changes

2 - 50 Copyright © 2014 Pearson Education, Inc. All rights reserved Private Colleges’ Student-to-Teacher Ratio  Typical ratio between 10 and 11.  Skewed right.  Outlier of 54 student-to-teacher ratio.  Large Variation.

2 - 51 Copyright © 2014 Pearson Education, Inc. All rights reserved Public Colleges’ Student-to-Teacher Ratio  Typical ratio between 16 and 20.  Generally symmetric.  Outlier of fractional student-to-teacher ratio.  Less Variation.

2 - 52 Copyright © 2014 Pearson Education, Inc. All rights reserved Comparing the Histograms It is much easier to describe the data when they are displayed using histograms compared to just the raw data table.

2 - 54 Copyright © 2014 Pearson Education, Inc. All rights reserved Eating Out for Students With Full Time Jobs vs. Part Time Jobs  Full time jobs: 5, 3, 4, 4, 4, 2, 1, 5, 6, 5, 6, 3, 3, 2, 4, 5, 2, 3, 7, 5, 5, 1,4, 6, 7  Part time jobs: 1, 1, 5, 1, 4, 2, 2, 3, 3, 2, 3, 2, 4, 2, 1, 2, 3, 2, 1, 3, 3, 2,4, 2, 1

2 - 55 Copyright © 2014 Pearson Education, Inc. All rights reserved Create a Dot Plot  Full time jobs: 5, 3, 4, 4, 4, 2, 1, 5, 6, 5, 6, 3, 3, 2, 4, 5, 2, 3, 7, 5, 5, 1,4, 6, 7  Part time jobs: 1, 1, 5, 1, 4, 2, 2, 3, 3, 2, 3, 2, 4, 2, 1, 2, 3, 2, 1, 3, 3, 2,4, 2, 1

2 - 56 Copyright © 2014 Pearson Education, Inc. All rights reserved Examine Shapes  Full time jobs: Relatively mound shaped.  Part time jobs: Slightly skewed right.

2 - 57 Copyright © 2014 Pearson Education, Inc. All rights reserved Examine Center  Full time jobs: Typically eat out 5 times per week  Part time jobs: Typically eat out 2 times per week

2 - 58 Copyright © 2014 Pearson Education, Inc. All rights reserved Examine Variation  Full time jobs: Larger Variation - from once to 7 times per week.  Part time jobs: Smaller variation – from once to 5 times per week.

2 - 59 Copyright © 2014 Pearson Education, Inc. All rights reserved Check for Outliers  Full time jobs: No gaps, so no clear outliers  Part time jobs: No gaps, so no clear outliers.

2 - 60 Copyright © 2014 Pearson Education, Inc. All rights reserved Summarize  The typical part time worker eats out less often compared to the typical full time worker. There is wider variation for the eating out by full time workers than by part time workers. The shape of the distribution for full time workers is approximately mound shaped, while it is slightly skewed right for part time workers.