Organizing Quantitative Data

Transcription

1 Organizing Quantitative Data MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2018

2 Objectives At the end of this lesson we will be able to: organize discrete data in tables, construct histograms of discrete data, organize continuous data in tables, construct histograms of continuous data, draw stem-and-leaf plots, draw dot plots, identify the shape of a distribution, draw time-series graphs.

3 Organizing Discrete Data If there are relatively few values of the variable, we may treat them the same as qualitative data. If there are many values of the variable, we create categories called classes using intervals of numbers.

4 Few Values of the Variable Example Construct a frequency and relative frequency distribution for the final exam scores of students in an earlier semester of MATH Remark: the grades range from 67 to 77 (only 11 different possible grades) so we treat the data as if it were qualitative.

5 Solution Grade Frequency Relative Frequency Total

6 Histogram Definition A histogram is constructed by drawing rectangles for each class of data. The height of each rectangle is the frequency or relative frequency of the class. The width of each rectangle is the same and the rectangles touch each other.

7 Example Construct a frequency histogram of the final exam grades presented earlier (repeated below for convenience)

8 Histogram of Frequencies Freq Grades

9 Histogram of Relative Frequencies Rel. Freq Grades

10 Continuous Data Continuous data must be organized into intervals of numbers called classes. The lower class limit of a class is the smallest value within the class. The upper class limit of a class is the largest value within the class. The class width is the difference between two consecutive lower class limits. A table is open ended if the first class has no lower class limit or the last class has no upper class limit. There is no best choice of class width. We usually pick a class width which produces 5 12 classes. class width maximum minimum number of classes

11 Example Consider the following data representing the length in minutes of final round tennis matches Construct a frequency and relative frequency table for the data with five categories.

12 Solution (1 of 2) It will be helpful if we start by sorting the data in ascending order

13 Solution (2 of 2) The minimum and maximum times are respectively 40.1 and 89.3 respectively. Thus if we choose the first lower class limit to be 40 and the class width to be 10, we can summarize the data as follows. Class Frequency Relative Frequency

14 Histogram of Continuous Data To create a histogram of the data we label the lower class limits on the horizontal axis and the class frequency (or relative frequency) on the vertical axis. Example Construct a histogram of the tennis match data using the frequencies just determined in the previous table.

15 Frequency Histogram Freq

16 Relative Frequency Histogram Rel. Freq Time

17 Stem-and-Leaf Plots 1. The stem of the graph will consist of the digits to the left of the right-most digit. The leaf of the graph will be the rightmost digit. 2. Write the stems in a vertical column in increasing order. Draw a vertical line to the right of the stems. 3. Write each leaf corresponding to the stems to the right of the vertical line. 4. Write the leaves in ascending order.

18 Example Round the tennis match times to the nearest whole minute and draw a stem-and-leaf plot. First round the data given earlier The stems are the tens digits of the data and the leaves are the ones digits of the data.

19 Solution

20 Splitting Stems We may use more than one stem for a class of data

21 Dot Plots A dot plot is drawn by placing each observation horizontally in increasing order and placing a dot above the observation each time it is observed. Example Draw a dot plot of the final exam grades presented earlier and repeated below for convenience

22 Solution 67

23 Shapes of Distributions We may describe variables through the shape of its histogram. uniform, frequency of each value of the variable is evenly spread across the values of the variable. bell-shaped, highest frequency occurs in the middle and frequencies tail off to the left and right. skewed right, tail to the right of the peak is longer than the tail to the left of the peak. skewed left, tail to the left of the peak is longer than the tail to the right of the peak.

24 Uniform Distribution

25 Bell-Shaped Distribution

26 Distribution Skewed Right

27 Distribution Skewed Left

28 Time-Series Graphs If the value of a variable is measured at different points in time, the data are referred to as time-series data. Definition A time-series plot is obtained by plotting the time in which a variable is measured on the horizontal axis and the corresponding value of the variable on the vertical axis. Line segments are then drawn connecting the points.

29 Example Draw a time-series plot of housing permits issued according to the following table. Housing Permits Year (in thousands)

30 Time-Series Plot Permits Year