DS4 Interpreting Sets of Data

Basic Concepts: Representing grouped data (or large sets of data) using frequency tables, histograms, stem-and-leaf plots, box-and-whisker plots, radar charts, two-way tables Estimating and comparing measures of spread and location Identifying outliers in data sets Describing shapes of distributions in terms of skewness, smoothness, general characteristics

To remind you…

Frequency histograms and Polygons

Cumulative Frequency Cumulative frequency is the frequency of the score plus the frequency of all the scores less than that score.

Measures of Spread Compare these two sets of data: Set A: 8, 9, 10, 11, 12 Set B: 0, 1, 10, 19, 20 Explain the differences using statistical language…

Using an ogive

Shape of frequency distributions The shape of frequency curves may be described in terms of smoothness, symmetry and number of modes. Graph A is a smooth curve, Graph B is not smooth. Graph C is unimodal (has one mode) and graph D is bimodal (has two modes). Graph E is symmetrical, Graph F is asymmetrical. Graphs which are not symmetrical are said to be skewed.

Skew If the longer tail of the graph is to the left, then the distribution is negatively skewed. This would occur, for example, if we graphed the distribution of the results of a very easy test. Most of the students would score high marks and only a few would score low marks. If the longer tail is to the right, then the distribution is positively skewed. This would occur for the distribution of the results of a very hard test.

Other shapes that occur regularly enough to be of importance are shown. This curve is described as bell-shaped. It occurs for many naturally occurring characteristics. This is called a J-shaped distribution because of its similarity to the shape of this letter. As the value of the variable increases so does the frequency of that variable. For a reverse J-shaped distribution, the value of the variable decreases as the frequency of occurrence increases. A U-shaped distribution is U-shaped. A uniform distribution has no mode.

Outliers An outlier is a score that is separated from the majority of the data. In this course, an outlier is defined as: QL − 1.5 × IQR or QU + 1.5 × IQR.

Displaying Data

Print master A Box-and-whisker plot uses the five-number summary (minimum, Q1, median, Q3, maximum). Two box-and-whisker plots on the same scale can help compare data sets.

Radar charts

Area charts An area chart is used to display and compare similar quantities. It consists of different ‘areas’, each representing a data set over a period of time. The thickness of the area indicates the size of the data. In January the rainfall of Town B is 15 and the rainfall of Town A is 10 (not 25). Furthermore, the area chart shows that during April both towns had the same rainfall (equal areas).

Comparison of summary statistics The selection and the use of the appropriate measure of location (mean or median) and measure of spread (range, interquartile range or standard deviation) depends on the nature of the data and the relative merits of each measure.

Two-way tables