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Contents
- 1 Collecting and Presenting Data
- 1.1 Introduction
- 1.1.1 Examples
- 1.1.2 Definitions
- 1.1.3 Surveys
- 1.2 Sampling
- 1.2.1 Simple Random Sampling
- 1.2.2 Stratified Sampling
- 1.2.3 Systematic Sampling
- 1.2.4 Multi-stage Sampling
- 1.2.5 Cluster Sampling
- 1.2.6 Judgemental sampling
- 1.2.7 Accessibility sampling
- 1.2.8 Quota Sampling
- 1.2.9 Sample Size
- 1.3 Frequency Tables
- 1.3.1 Frequency Tables
- 1.3.2 Continuous Data Frequency Tables
- 1.4 Exercises
- 2 Graphical methods for presenting data
- 2.1 Introduction
- 2.2 Stem and Leaf plots
- 2.3 Bar Charts
- 2.4 Multiple Bar Charts
- 2.5 Histograms
- 2.6 Exercises
- 3 More graphical methods for presenting data
- 3.1 Introduction
- 3.2 Percentage Relative Frequency Histograms
- 3.3 Relative Frequency Polygons
- 3.4 Cumulative Frequency Polygons (Ogive)
- 3.5 Pie Charts
- 3.6 Time Series Plots
- 3.7 Scatter Plots
- 3.8 Exercises
- 4 Numerical summaries for data
- 4.1 Introduction
- 4.2 Mathematical notation
- 4.3 Measures of Location
- 4.3.1 The Mean
- 4.3.2 The Median
- 4.3.3 The Mode
- 4.4 Measures of Spread
- 4.4.1 The Range
- 4.4.2 The Inter-Quartile Range
- 4.4.3 The Sample Variance and Standard Deviation
- 4.5 Summary statistics in MINITAB
- 4.6 Box and Whisker Plots
- 4.7 Exercises
- 5 Introduction to Probability
- 5.1 Introduction
- 5.2 How do we measure Probability?
- 5.2.1 Classical
- 5.2.2 Frequentist
- 5.2.3 Subjective/Bayesian
- 5.3 Laws of Probability
- 5.3.1 Multiplication Law
- 5.3.2 Addition Law
- 5.3.3 Example
- 5.4 Exercises
- 6 Decision Making using Probability
- 6.1 Conditional Probability
- 6.2 Tree Diagrams
- 6.3 Expected Monetary Value and Probability Trees
- 6.4 Exercises
- 7 Discrete Probability Models
- 7.1 Introduction
- 7.2 Permutations and Combinations
- 7.2.1 Permutations
- 7.2.2 Combinations
- 7.3 Probability Distributions
- 7.3.1 Expectation and the population mean
- 7.3.2 Population variance and standard deviation
- 7.4 The Binomial Distribution
- 7.5 The Poisson Distribution
- 7.6 Exercises
- 8 Continuous Probability Models
- 8.1 Introduction
- 8.2 The Uniform Distribution
- 8.3 The Exponential Distribution
- 8.4 The Normal Distribution
- 8.4.1 Notation
- 8.4.2 Probability calculations and the standard normal distribution
- 8.5 Exercises
Chapter 3
More graphical methods for presenting data
3.1 Introduction
We have seen some basic ways in which we might present data graphically. These methods will often provide the mainstay of business presentations. There are, however, other techniques which are useful and offer advantages in some applications over histograms and bar charts.
3.2 Percentage Relative Frequency Histograms
When we produced frequency tables in Chapter 2, we included a column for percentage relative frequency. This contained values for the frequency of each group, relative to the overall sample size, expressed as a percentage. Recall the data on service time (in seconds) for calls to a credit card service centre:
214.8412 220.6484 216.7294 195.1217 211. 195.8980 201.1724 185.8529 183.4600 178. 196.3321 199.7596 206.7053 203.8093 203. 200.8080 201.3215 205.6930 181.6718 201. 180.2062 193.3125 188.2127 199.9597 204. 198.3838 193.1742 204.0352 197.2206 193. 205.5048 217.5945 208.8684 197.7658 212. 209.9000 197.6215 204.9101 203.1654 192. 208.9901 202.0090 195.0241 192.7098 219. 208.8920 200.7965 191.9784 188.8587 206.
A percentage relative frequency table for these data is
- Click on OK and again on OK.
This produces the following histogram:
Note that the y-axis now contains the relative percentages rather than the frequencies. You might well ask why we would want to do this? These percentage relative frequency histograms are useful when comparing two samples that have different numbers of observations. If one sample were larger than the other then a frequency histogram would show a difference simply because of the larger number of observations. Looking at percentages removes this difference and enables us to look at relative differences. It is really just a matter of making the vertical scales comparable. In the following graph there are data from two groups and four times as many data points for one group as the other. The left-hand plot shows an ordinary histogram and it is clear that the
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comparison between groups is masked by the quite different sample sizes. The right-hand plot shows a histogram based on (percentage) relative frequencies and this enables a much more direct comparison of the distributions in the two groups.
Overlaying histograms on the same graph can sometimes not produce such a clear picture, partic- ularly if the values in both groups are close or overlap one another significantly.
3.3 Relative Frequency Polygons
These are a natural extension of the relative frequency histogram. They differ in that, rather than drawing bars, each class is represented by one point and these are joined together by straight lines. The method is similar to that for producing a histogram.
- Produce a percentage relative frequency table.
- Draw the axes
- The x-axis needs to contain the full range of the classes used.
- The y-axis needs to range from 0 to the maximum percentage relative frequency.
- Plot points, pick the mid point of the class interval on the x-axis and go up until you reach the appropriate percentage value on the y-axis and mark the point. Do this for each class.
- Join the points together with straight lines.
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- Add a title by clicking on Labels... etc.
- Click on OK.
These instructions produce the graph
These percentage relative frequency polygons are of most use however for comparison between two samples. Consider the following data on gross weekly income collected from two sites in Newcastle. Let us suppose that many more responses were collected in Jesmond so that a di- rect comparison of the frequencies using a standard histogram is not appropriate. Instead we use relative frequencies.
Weekly Income ( £ ) West Road (%) Jesmond Road (%) 0 ≤ income < 100 9.3 0. 100 ≤ income < 200 26.2 0. 200 ≤ income < 300 21.3 4. 300 ≤ income < 400 17.3 16. 400 ≤ income < 500 11.3 29. 500 ≤ income < 600 6.0 22. 600 ≤ income < 700 4.0 17. 700 ≤ income < 800 3.3 4. 800 ≤ income < 900 1.3 2. 900 ≤ income < 1000 0.0 2.
We can produce a graph containing polygons for both locations using MINITAB instructions very similar to those above:
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- Place the data in the worksheet using column C1 for the mid-points, column C2 for the percentage relative frequencies and column C3 for the site where the data were taken.
- Graph > Scatterplot...
- Select the With Connect and Groups option and click on OK
- Enter the column with the percentage frequencies (C2) under Y variables and the col- umn with the midpoints (C1) under X variables. Also enter the Site column (C3) in the box for Categorical variables for grouping
- Add a title by clicking on Labels... etc.
- Click on OK.
The polygon produced looks like
