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Module 4 Graphs and Plots

Learning Objectives

 Determine the utility of using graphs to represent data

 Determine methods to graph continuous data

 Identify symmetry, skew, and kurtosis in a distribution

 Determine methods to graph discrete data

Module Summary

 Although frequency tables provide a neat method for organizing data, graphs can be an even more effective method for presenting information. Information that can be obtained from a graph includes the dispersion, clustering, and location of the majority of scores.

 When creating a graph, the traditional rules are that the X-axis (abscissa) represents the intervals of the measured variable and the Y-axis (ordinate) represents the frequency of scores at each interval. In situations where there is a large frequency of cases for a particular score interval, you can divide the interval on the X-axis.

 There are some rules for creating a graph, which are as follows: (1) The Y-axis should be 3/4 the size of the X-axis. (2) With large data sets, you can collapse intervals on the X-axis so that there are at least 5 intervals but not more than 12 intervals. (3) Each interval on the X-axis must be equal to the others. (4) The Y-axis must be continuous. (5) The axes should not stretch or compress the data.

 Histograms are graphs for continuous data that indicate the frequency of a particular score by bars. The bars touch one another to indicate that a score could fall between the intervals on the X-axis.

 Frequency curves are alternatives to the histogram. A frequency curve is drawn by first creating a histogram and then connecting the midpoints of the adjacent bars with solid lines. This provides you with a visual representation of how your data are distributed, as it allows you to determine easily whether the data are clustered around a specific score or if they are spread out, and if they appear heavily lopsided or symmetrical.

 Frequency curves can take on multiple shapes. One of these shapes is the normal curve, which appears bell-shaped. This means that it is symmetrical in shape, with approximately half the scores falling above the peak (middle score) and half below. The other shapes involve skew, which refers to a lopsided distribution. Skew is produced by having a greater concentration of scores at either the upper or the lower end of the distribution. Skew is named by the direction of the long tail. If there are a lot of high scores, the distribution is said to be negatively skewed, as there is a long tail on the left. If there are a lot of low scores, the distribution is said to be positively skewed, as there is a long tail on the right.

 Boxplots or Box-and-whisker plots can be used to examine the distributions of two or more variables. A boxplot consists of a box and two whiskers. The box represents the middle 50% of a distribution. The top whisker represents the upper 25% of the distribution and the bottom whisker represents the lower 25%. The bottom whisker extends from the bottom of the box to the lowest value of the distribution, and the top whisker extends from the top of the box to the highest value of the distribution. The line running through the box represents the median (the middle value) of the distribution.

 Kurtosis refers to the amount of scores that are in the middle of the distribution. Distributions with a lot of scores in the center are referred to as leptokurtic. Alternatively, distributions with few scores in the center are referred to as platykurtic.

 The other two general shapes of the frequency curve are bimodal and uniform. Bimodal distributions have more than one peak. Uniform distributions are uniform in the spread of responses, which means the frequency of all the scores is the same.

 When graphing nominal data, it is more appropriate to use a bar graph or a pie graph, as these methods express the discrete nature of nominal variables. Bar graphs appear similar to histograms except that the bars are separated to indicate that a score could not fall between adjacent categories. Pie graphs place the nominal data in a circle, with slices to represent the different categories. The size of each slice indicates the amount of participants or cases in that particular category.

Computational Exercises

1 Create a histogram that accurately portrays these data using 1-point intervals: 47, 48, 48, 49, 49, 49 50, 50, 50, 51, 51, 52.

2 Create a frequency curve for these data using 1-point intervals.

3 Describe the shape of this distribution in terms of skewness and kurtosis.

4 If you were to create a positively skewed distribution, where would you need to add scores? A negatively skewed distribution? What would you need to do to make the distribution bimodal? Uniform?

5 State how you would expect the distributions for the following variables to appear:Time it takes a random group of people to run a mileAmount of money earned in the first year after completing collegeAge of football playersHappiness of guests during a wedding

6 Which would be the appropriate methods for graphing the following data?Number of trophies won by tennis playersLength of time in therapy for obsessive compulsive disorderNumber of words spoken by 3-year-oldsThe profit margins of six different companies for the past year

Computational Answers

3 The data are symmetrical, and there is no evidence of kurtosis.

4 To create a positively skewed distribution, you would need to add a large amount of scores below 47 or a few scores greater than 52. To create a negatively skewed distribution, you would need to add a large amount of scores above 52 or a few scores less than 47. To create a bimodal distribution, you would need to add a second group of scores with a similar peak. To create a uniform distribution, you would need to make it so all scores had an equal frequency.

5 Expect a positively skewed distribution, as there will be certain people who can run it quite quickly, but the majority will take a long time.Symmetrically distributed, as the majority will earn approximately the same amount, with fewer earning higher or lower.Positively skewed, with the majority of players being younger.Negatively skewed, with the majority of guests very happy.

6 Bar graph, pie chartHistogram, frequency curveFrequency curve, histogramFrequency curve, histogram

True/False Questions

1 A score’s percentile rank depends on the number of scores there are at that interval and the upper and lower limits of the score.

2 Pie charts and bar charts are excellent methods for graphing nominal data.

3 Boxplots are useful for comparing distributions of multiple variables.

4 A distribution with a lot of high scores and very few low scores would be considered negatively skewed.

5 In a symmetrical distribution, the majority of the scores are above the midpoint.

6 Kurtosis refers to the height of the middle scores of a distribution.

True/False Answers

1 True

2 True

3 True

4 True

5 False

6 True

Short-Answer Questions

1 What pieces of information can you obtain from a boxplot (or box-and-whisker plot)?

2 Why are the bars on a histogram connected?

3 What is the difference between a positively skewed distribution and a negatively skewed distribution?

4 You notice that the scores in a recent survey for how much viewers like a new TV show have a uniform distribution. What does this mean about the responses in your sample?

5 What is kurtosis, and what are its two forms?

6 What do the slices on a pie graph represent?

Answers

1 From a boxplot, we can obtain the median of the distribution, where the middle 50% of scores fall, and the maximum and minimum values.

2 This indicates that the values could fall anywhere between the intervals.

3 A positively skewed distribution has a large amount of cases at the lower end of the scale and a few at the upper end. A negatively skewed distribution has a large amount of cases at the upper end of the scale and few toward the lower end.

4 Each score (interval) on your survey received an equal amount of responses.

5 Kurtosis refers to having either more or fewer scores in the center of a distribution, relative to a normal distribution. A distribution with more scores at the center is referred to as leptokurtic, and one with fewer is referred to as platykurtic.

6 The proportion of cases falling in each category. Pie charts are used for nominal data.

Multiple-Choice Questions

1 A pizza restaurant finds that a lot of customers appeared to eat a less number of pizza slices, and a large portion also appeared to eat more number of slices. How should the restaurant expect the distribution to look?Positively skewedNegatively skewedBimodalSymmetrical

2 The X-axis is sometimes referred to as the ______.abscissaordinateplatykurticleptokurtic

3 In a study examining the rate at which a person can mentally rotate an object, the results indicate that the majority of people take 15 seconds, with an equal amount of participants falling above and below this time. You could expect the relative frequency curve of these scores to appearsymmetric.positively skewed.negatively skewed.bimodal.

4 What would be the best method to graph data obtained for the question of “What is your favorite type of ice cream?”HistogramRelative frequency curveBar graphBoxplot

5 Bar and pie charts are best used for which type of variables?ContinuousOrdinalNominalRatio

6 You administer an anxiety measure to a group of 30 participants with different severity of anxiety symptoms. Two participants obtain each possible score. How will this distribution appear?UniformSymmetricalPositively skewedNegatively skewed

7 A common symptom of depression is a lack of desire to do things that are entertaining. If you were to ask a sample of 100 individuals with depression how many fun activities they have done in the past week, how would you expect the distribution to appear?UniformSymmetricalPositively skewedNegatively skewed

8 You are interested in determining how moviegoers rated a new film on opening day, on a scale of 1 to 8. After collecting data, you discover that most people in the sample provided ratings of either 4 or 5. How would you expect this distribution to appear?Positively skewedNegatively skewedLeptokurticPlatykurtic

Multiple-Choice Answers

1 C

2 A

3 A

4 C

5 C

6 A

7 C

8 C

Module Quiz

1 Create a histogram for the following data:1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 8, 9.What shape best describes this distribution?

2 You are working at an advertising firm and want to determine the interest of a focus group in a new product. If the scale ranges from 0 (no interest) to 10 (great interest), what type of distribution would be most preferable?

3 After giving his class a quiz worth 10 points (1–10 scale), your professor notices that the distribution was negatively skewed. What conclusion should the professor draw from this information?

4 What types of variables are histograms best used for?

5 What does the height of a bar in a histogram represent?

Quiz Answers


1 Positively skewed.

2 Negatively skewed.

3 The test was potentially too easy, he may need to add more challenging questions next time he gives a quiz.

4 Continuous variables.

5 The frequency of scores in that particular grouping or category.

Student Study Guide to Accompany Statistics Alive!

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