Student Study Guide to Accompany Statistics Alive!. Wendy J. Steinberg

       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

       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.

      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

      1 

      2 

      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

      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.

      1 True

      2 True

      3 True

      4 True

      5 False

      6 True

      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