Chapter 7: Statistics Describing Data. Chapter 7: Statistics Describing Data 1 / 27
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1 Chapter 7: Statistics Describing Data Chapter 7: Statistics Describing Data 1 / 27
2 Categorical Data Four ways to display categorical data: 1 Frequency and Relative Frequency Table 2 Bar graph (Pareto chart) 3 Pie chart 4 Pictogram Chapter 7: Statistics Describing Data 2 / 27
3 Frequency Table Definition (Frequency Table) A frequency table is a table with two columns. One column lists the categories, and another for the frequencies with which the items in the categories occur (how many items fit into each category). Favorite colors in a class of Kindergarteners: Red Red Yellow Green Blue Blue Blue Blue Pink Pink Pink Purple Color Frequency Red 2 Yellow 1 Green 1 Blue 4 Pink 3 Purple 1 Chapter 7: Statistics Describing Data 3 / 27
4 Relative Frequency Table Definition (Relative Frequency Table) A relative frequency table is a frequency table with a column of fractions or percents describing the relative frequency of each category. You roll a die 25 times. The rolls are: 4,4,2,2,1,6,6,6,5,1,4,2,1,4,6,5,5,5,3,2,1,2,2,4,1. Roll Frequency Relative Frequency Chapter 7: Statistics Describing Data 4 / 27
5 Bar Graphs Definition (Bar graph) A bar graph is a graph that displays a bar for each category with the length of each bar indicating the frequency of that category. Definition (Pareto chart) A Pareto chart is a bar graph ordered from highest to lowest frequency Chapter 7: Statistics Describing Data 5 / 27
6 Pareto Chart Example Frequency (%) Green Red Black White Blue Grey Vehicle color involved in total-loss collision Chapter 7: Statistics Describing Data 6 / 27
7 Pie Chart Definition (Pie Chart) A pie chart is a circle with wedges cut of varying sizes marked out like slices of pie or pizza. The relative sizes of the wedges correspond to the relative frequencies of the categories. Chrome 46.6 % Internet Explorer 24.6 % 2.0 % % 20.4 % Other Opera Safari Firefox Chapter 7: Statistics Describing Data 7 / 27
8 Pictogram Definition (Pictogram) A pictogram is a statistical graphic in which the size of the picture is intended to represent the frequencies or size of the values being represented. Chapter 7: Statistics Describing Data 8 / 27
9 Quantitative Data Graphical Summaries of Quantitative Data: 1 Histogram 2 Frequency Polygon Chapter 7: Statistics Describing Data 9 / 27
10 Histogram Definition (Histogram) A histogram is graph that displays a rectangle for each numerical class interval with the height of each rectangle indicating the frequency of values in the interval. A histogram is similar to a bar graph, but the horizontal axis is a number line. All class intervals must be an equal width. Interval Frequency Frequency Weights (pounds) Chapter 7: Statistics Describing Data 10 / 27
11 Frequency Polygon Definition (Frequency polygon) A frequency polygon is similar to a histogram, but instead of drawing a bar, a point is placed in the midpoint of each interval at height equal to the frequency. Typically the points are connected with straight lines to emphasize the distribution of the data. 40 Frequency Heights (in) Chapter 7: Statistics Describing Data 11 / 27
12 Measures of Central Tendency Definition (Mean) The mean of a set of data is the sum of the data values divided by the number of values. Definition (Median) The median of a set of data is the value in the middle when the data is in order Definition (Mode) The mode is the element of the data set that occurs most frequently. Chapter 7: Statistics Describing Data 12 / 27
13 Small Example 3,5,5,6,7,9 Mean = = Median = = 5.5 Mode = 5 Chapter 7: Statistics Describing Data 13 / 27
14 Large Example Data value Frequency Mean = = = = Chapter 7: Statistics Describing Data 14 / 27
15 Large Example Data value Frequency Median is the average of the 100th and 101st data values. Median = 18 Mode = 18 Chapter 7: Statistics Describing Data 15 / 27
16 Measures of Spread Definition (Range) The range is the difference between the maximum value and the minimum value of the data set. Definition (Standard deviation) The standard deviation is a measure of variation based on measuring how far each data value deviates, or is different, from the mean. Definition (Quartiles) Quartiles are values that divide the data in quarters. The first quartile (Q1) is the value so that 25% of the data values are below it; the third quartile (Q3) is the value so that 75% of the data values are below it. The second quartile is the same as the median, since the median is the value so that 50% of the data values are below it. Chapter 7: Statistics Describing Data 16 / 27
17 Example 1,2,6,6,7,9,18 Range = 18 1 = 17 Standard Deviation: Mean = = 49 7 = 7 Data value Deviation Deviation squared = Chapter 7: Statistics Describing Data 17 / 27
18 Example 1,2,6,6,7,9,18 Range = 18 1 = 17 Standard Deviation: Mean = = 49 7 = 7 Data value Deviation Deviation squared = = Chapter 7: Statistics Describing Data 18 / 27
19 Example 1,2,6,6,7,9,18 Range = 18 1 = 17 Standard Deviation: Mean = = 49 7 = 7 Data value Deviation Deviation squared = = = Chapter 7: Statistics Describing Data 19 / 27
20 Example 1,2,6,6,7,9,18 Range = 18 1 = 17 Standard Deviation: Mean = = 49 7 = 7 Data value Deviation Deviation squared = = = = Chapter 7: Statistics Describing Data 20 / 27
21 Example 1,2,6,6,7,9,18 Range = 18 1 = 17 Standard Deviation: Mean = = 49 7 = 7 Data value Deviation Deviation squared = = = = = Chapter 7: Statistics Describing Data 21 / 27
22 Example 1,2,6,6,7,9,18 Range = 18 1 = 17 Standard Deviation: Mean = = 49 7 = 7 Data value Deviation Deviation squared = = = = = = Chapter 7: Statistics Describing Data 22 / 27
23 Example 1,2,6,6,7,9,18 Range = 18 1 = 17 Standard Deviation: Mean = = 49 7 = 7 Data value Deviation Deviation squared = = = = = = = = = 5.6 Chapter 7: Statistics Describing Data 23 / 27
24 Standard Deviation 1 Calculate the mean. 2 Calculate the deviation from the mean for each data value. 3 Square each deviation. 4 Add the squared deviations. 5 Divide by one less than the number of data values. 6 Take the square root. Chapter 7: Statistics Describing Data 24 / 27
25 Example 1,2,6,6,7,9,18 Range = 18 1 = 17 Standard Deviation=5.6 First Quartile, Q 1 = 2 Third Quartile, Q 3 = 9 Chapter 7: Statistics Describing Data 25 / 27
26 5-number summary Definition (Five number summary) The five number summary takes this form: Minimum, Q1, Median, Q3, Maximum 5-number summary: 1,2,6,9,18 1,2,6,6,7,9,18 Chapter 7: Statistics Describing Data 26 / 27
27 Box plot Definition (Box plot) A box plot is a graphical representation of a five-number summary. To create a box plot, a number line is first drawn. A box is drawn from the first quartile to the third quartile, and a line is drawn through the box at the median. Whiskers are extended out to the minimum and maximum values Chapter 7: Statistics Describing Data 27 / 27
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