2.1: Frequency Distributions and Their Graphs

Transcription

1 2.1: Frequency Distributions and Their Graphs Frequency Distribution - way to display data that has many entries - table that shows classes or intervals of data entries and the number of entries in each class - the frequency, f, of a class is the number of data entries in the class Constructing a frequency distribution 1. Decide on the number of classes to include 2. Find class width. Round up to the next convenient number. Max Min Class width Number of classes( Given) 3. Find class limits. Use the minimum data entry and the lower limit of the first class. To find the remaining lower limit, add the class width to the lower limit of the preceding class. 4. Use tally marks to total the entries for each class. 5. Count the tally marks to find the frequency for each class. For Example.

2 Example 1: Construct a frequency distribution using the ages of the residents of Akhiok given below. Use six classes. 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 9, 10, 10, 10, 11, 11, 11, 12, 12, 13, 15, 16, 16, 17, 17, 21, 21, 22, 23, 24, 25, 25, 26, 27, 27, 28, 28, 29, 30, 31, 31, 32, 32, 33, 33, 34, 36, 39, 41, 42, 45, 46, 47, 48, 49, 50, 50, 51, 52, 53, 54, 55, 56, 60, 61 Definitions Midpoint sum of the lower and upper limits of each class divided by two. After finding the first midpoint, you can find the remaining midpoints by adding the class width to the previous midpoint. Relative Frequency portion or percent of the data that falls in that class. Divide the frequency, f, of the class by the sample size, n. Cumulative Frequency sum of the frequency for that class and all previous classes. The cumulative frequency of the last class is equal to the sample size.

3 Example 2: Using the frequency distribution constructed in Example1, find the midpoint, relative frequency, and cumulative frequency for each class. Identify any patterns and characteristics. LB Class Limits UB Midpoint f Cumulative f Relative f Graphs of Frequency Distributions A frequency histogram is a bar graph that represents the frequency distribution of a data set. 1. The horizontal scale is quantitative and measures the data values. 2. The vertical scale measures the frequencies of the classes 3. Consecutive bars must touch. Therefore, bars must begin and end at class boundaries instead of class limits. You can mark the horizontal scale using the class boundaries or midpoints.

4 A frequency polygon is a line graph that emphasizes the continuous changes in frequencies. Use midpoints.

5 Relative Frequency Histogram-vertical scale measure relative frequencies. Cumulative Frequency Graph (Ogive) a line graph that displays the cumulative frequency of each class at its upper class boundary. The upper boundaries are marked on the horizontal axis and the cumulative frequencies are marked on the vertical axis.

6 2.2: More Graphs and Displays Graphing Quantitative Data Sets Stem-and-Leaf Plot newer way of displaying data. Construct 2 stem and leaf plots. For the first, use the key 1 55 = 155. For the second, use the key 15 5 = 155. Dot Plot each data entry is plotted, using a point, above a horizontal axis. Data from Example 1 turned into a dot plot: Are there any outliers? Circle the outliers. Should they be removed? Why or why not?

7 Graphing Qualitative Data Sets Pie Chart circle graph. Steps to constructing a pie chart: 1. Find the relative frequency of each data entry. 2. Multiply each relative frequency by 360 to find the appropriate angle measures. 3. Construct pie chart and be sure to label appropriately. (Label using percents.) Example 4: The numbers of motor vehicle occupants killed in crashes in 1995 are shown in the table. Use a pie chart to organize the data.

8 Pareto Chart Vertical bar graph in which the height of the bar represents frequency or relative frequency.

9 Graphing Paired Data Sets Scatter Plots ordered pairs are graphed as points in a coordinate plane. Example 6: The lengths of employment and the salaries of 10 employees are listed in the table at the left. Graph the data using a scatter plot. Length of Employment (in Salary (in dollars) years) 5 32, , , , , , , , , ,000

10 Time Series Chart composed of entries taken at regular intervals over a period of time. Example 7: Use the table to construct a time series chart for a subscribers average local monthly cellular telephone bill for the years 1987 through Use a broken x-axis. Year Average Bill (in dollars)

11 2.3: Measures of Central Tendency Measures of Central Tendency represents a typical, or central, entry of a data set. Median middle data entry when the data set is sorted in ascending or descending order. If the data set has an even # of entries, the median is the mean of the two middle data entries. Mode data entry that occurs the most often. If no entry is repeated, the data set has no mode. If there is a tie, then the data set is bimodal meaning it has 2 modes! Example 1: Find the sample mean, median and mode for the following data set Mean = Median = Mode = Are there any outliers (values that look like they do not belong)? Remove them and recalculate. Mean = Mode = Median = Is the data set: Without a Mode? Modal? Bimodal?

12 Weighted Mean and Mean of Grouped Data data sets that contain entries that have a greater effect on the mean than do other entries. (For example: the way your grade is calculated in this class). Weighted mean mean of a data set whose entries have varying weights. xw x, where w is the weight of each entry. w Example 2: Your grade in this class is determined from five scores: 45% from your test mean, 20% from final exam, 15% from quizzes and 20% from homework. Your scores are 86 (test), 96 (final exam), 82 (quizzes), and 100 (homework). What is the weighted mean (your average in the class) of your scores? Source Score, x Weight, w (xw) Test Final Quiz Homework w 1 Add the column. xw Add the column. Mean of a frequency distribution - the frequency of a class. x xf n, where x is the midpoint of a class and f is Example 3: Use the following frequency distribution to determine the mean. Height Frequency, f Midpoint, x xf

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14 2.4: Measures of Variation Definitions and Formulas Range difference between the maximum and minimum entries in a data set Deviation The deviation of an entry x in a population set is the difference between the entry and the population mean. Deviation of xx * For any data set, the sum of the deviations is zero, so finding the average of the deviations wouldn t make sense. We use population variance. Population Variance the mean of the squares of the deviations. 2 x 2 Population Variance, where N is the number of entries. N Population Standard Deviation allows you to return to the original unit of the data set. Population standard deviation x 2 N Example 1: Corporation A has hired 10 graduates. The starting salaries for the 10 graduates at corporation A are as follows. Starting Salaries for corporation A (in thousands of dollars) Salary Question 1: What is the range? Question 2: What is the population variance and standard deviation? Data Entries Mean Deviation x x x 2 = x 2 x 2 5 This number is your variance N x 2 N 16 This number is your s.d.

15 * Sometimes information about a population is unknown so variance and standard deviation must be estimated using a sample. Sample variance s 2 Sample standard deviation x x 2 n 1 2 s s x x 2 n 1 Question 3: What is the sample variance and standard deviation? This part is easy go back to step 4, and divide that number by n-1 instead. Recalculate from there. Example 2: Now complete the previous problem using your calculator. Calculator stuff Stat Calc 1-VAR STATS x x x 2 sx x n

16 Empirical Rule o Helps you see how valuable the standard deviation can be as a measure of variation. o For data with a bell-shaped distribution, the standard deviation has the following characteristics. 1. About 68% of the data set lies within 1 standard deviation of the mean. 2. About 95% of the data set lies within 2 standard deviations of the mean. 3. About 99.7% of the data set lies within 3 standard deviations of the mean.

17 Example 4: Heights of American Women Sample mean = 64 Sample standard deviation = Estimate the percent of heights that are between 64 and 69.5 inches. 2. Estimate the percent of heights that are below inches. 3. Estimate the percent of heights that are above Estimate the percent of heights that are below and above Between what two heights do 68% of all American women fall between? 6. Between what two heights do 95% of all American women fall between? 7. Between what two heights do 99.7% of all American women fall between?

18 2.5: Measures of Position Fractiles are numbers that divide an ordered data set into equal parts. o Quartiles divide a data set into four equal parts. o Deciles divide a data set into 10 equal parts. o Percentiles divide a data set into 100 equal parts. Example 1: Find the first, second, and third quartiles of the following data set Interquartile Range (IQR) Q3 Q1, this tells how much the middle half of a data set varies. Find the IQR for example 1: Example 2 Using the TI-83 graphing calculator find Q 1, Q 2, and Q 3 of the following data set

19 . Example 3: Using a calculator find the 5-number summary for example 1 and draw a box-and-whisker plot. 1 VAR STAT Outliers Numbers in a data set that fall 1.5IQR away from Q 1 or Q 3. Example 4: Are there any outliers for the data set in Example 1? If so, what are they? Other TILES!!!!

20 Now another idea.

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