Chapter 2 Descriptive Statistics

Upper Class Limits Lower Class Limits Frequencies Frequency Distributions A frequency distribution is a table that shows classes or intervals of data with a count of the number in each class. The frequency f of a class is the number of data points in the class.

5 – 1 = 4 9 – 5 = 4 13 – 9 = 4 17 – 13 = 4 Frequency Distributions The class width is the distance between lower (or upper) limits of consecutive classes. The class width is 4. The range is the difference between the maximum and minimum data entries.

Constructing a Frequency Distribution • Guidelines • Decide on the number of classes to include. The number of classes should be between 5 and 20; otherwise, it may be difficult to detect any patterns. • Find the class width as follows. Determine the range of the data, divide the range by the number of classes, and round up to the next convenient number. • Find the class limits. You can use the minimum entry as the lower limit of the first class. To find the remaining lower limits, add the class width to the lower limit of the preceding class. Then find the upper class limits. • Make a tally mark for each data entry in the row of the appropriate class. • Count the tally marks to find the total frequency f for each class.

Constructing a Frequency Distribution Example: The following data represents the ages of 30 students in a statistics class. Construct a frequency distribution that has five classes. Ages of Students Continued.

36 5 Constructing a Frequency Distribution Example continued: 1. The number of classes (5) is stated in the problem. 2. The minimum data entry is 18 and maximum entry is 54, so the range is 36. Divide the range by the number of classes to find the class width. Round up to 8. = 7.2 Class width = Continued.

Constructing a Frequency Distribution Example continued: 3. The minimum data entry of 18 may be used for the lower limit of the first class. To find the lower class limits of the remaining classes, add the width (8) to each lower limit. The lower class limits are 18, 26, 34, 42, and 50. The upper class limits are 25, 33, 41, 49, and 57. 4. Make a tally mark for each data entry in the appropriate class. 5. The number of tally marks for a class is the frequency for that class. Continued.

Number of students Ages • Class • Tally • Frequency, f Check that the sum equals the number in the sample. Constructing a Frequency Distribution Example continued: Ages of Students • 18 – 25 • 13 • 26 – 33 • 8 • 34 – 41 • 4 • 42 – 49 • 3 • 50 – 57 • 2

(Lower class limit) + (Upper class limit) 2 • Class • Frequency, f • Midpoint • 1 – 4 • 4 Midpoint The midpoint of a class is the sum of the lower and upper limits of the class divided by two. The midpoint is sometimes called the class mark. Midpoint = • 2.5 Midpoint =

Ages of Students • Class • Frequency, f • Midpoint • 18 – 25 • 13 • 26 – 33 • 8 • 34 – 41 • 4 • 42 – 49 • 3 • 50 – 57 • 2 Midpoint Example: Find the midpoints for the “Ages of Students”frequency distribution. 18 + 25 = 43 • 21.5 43 2 = 21.5 • 29.5 • 37.5 • 45.5 • 53.5

Class frequency Sample size Relative Frequency The relative frequency of a class is the portion or percentage of the data that falls in that class. To find the relative frequency of a class, divide the frequency f by the sample size n. Relative frequency = • 0.222 Relative frequency

Class • Frequency, f Portion of students • Relative Frequency • 13 • 8 • 18 – 25 • 4 • 26 – 33 • 3 • 34 – 41 • 2 • 42 – 49 • 50 – 57 Relative Frequency Example: Find the relative frequencies for the “Ages of Students”frequency distribution. • 0.433 • 0.267 • 0.133 • 0.1 • 0.067

Ages of Students • Class • Frequency, f • Cumulative Frequency Total number of students • 13 • 8 • 18 – 25 • 4 • 26 – 33 • 3 • 34 – 41 • 50 – 57 • 2 • 42 – 49 Cumulative Frequency The cumulative frequency of a class is the sum of the frequency for that class and all the previous classes. • 13 • 21 + • 25 + + • 28 + • 30

Frequency Histogram A frequency histogram is a bar graph that represents the frequency distribution of a data set. • The horizontal scale is quantitative and measures the data values. • The vertical scale measures the frequencies of the classes. • Consecutive bars must touch. Class boundaries are the numbers that separate the classes without forming gaps between them. The horizontal scale of a histogram can be marked with either the class boundaries or the midpoints.

Ages of Students • Class • Frequency, f • Class Boundaries • 13 • 8 • 18 – 25 • 4 • 26 – 33 • 3 • 34 – 41 • 2 • 42 – 49 • 50 – 57 Class Boundaries Example: Find the class boundaries for the “Ages of Students”frequency distribution. The distance from the upper limit of the first class to the lower limit of the second class is 1. • 17.5 25.5 • 25.5 33.5 • 33.5 41.5 • 41.5 49.5 Halfthisdistanceis 0.5. • 49.5 57.5

14 Ages of Students 13 12 10 8 8 f 6 4 4 3 2 2 0 17.5 25.5 33.5 41.5 49.5 57.5 Broken axis Age (in years) Frequency Histogram Example: Draw a frequency histogram for the “Ages of Students” frequency distribution. Use the class boundaries.

Ages of Students 14 12 Line is extended to the x-axis. 10 8 f 6 4 2 0 13.5 21.5 29.5 37.5 45.5 53.5 61.5 Midpoints Broken axis Age (in years) Frequency Polygon A frequency polygon is a line graph that emphasizes the continuous change in frequencies.

0.5 Ages of Students Relative frequency (portion of students) 0.4 0.3 0.2 0.1 0 17.5 25.5 33.5 41.5 49.5 57.5 Age (in years) Relative Frequency Histogram A relative frequency histogram has the same shape and the same horizontal scale as the corresponding frequency histogram. 0.433 0.267 0.133 0.1 0.067

Ages of Students 30 24 The graph ends at the upper boundary of the last class. 18 Cumulative frequency (portion of students) 12 6 0 17.5 25.5 33.5 41.5 49.5 57.5 Age (in years) Cumulative Frequency Graph A cumulative frequency graph or ogive, is a line graph that displays the cumulative frequency of each class at its upper class boundary.

More Graphs and Displays § 2.2

Stem-and-Leaf Plot In a stem-and-leaf plot, each number is separated into a stem (usually the entry’s leftmost digits) and a leaf (usually the rightmost digit). This is an example of exploratory data analysis. Example: The following data represents the ages of 30 students in a statistics class. Display the data in a stem-and-leaf plot. Ages of Students Continued.

Most of the values lie between 20 and 39. Stem-and-Leaf Plot Ages of Students Key: 1|8 = 18 8 8 8 9 9 9 1 2 3 4 5 0 0 1 1 1 2 4 7 9 9 0 0 2 2 3 4 7 8 9 4 6 9 1 4 This graph allows us to see the shape of the data as well as the actual values.

Stem-and-Leaf Plot Example: Construct a stem-and-leaf plot that has two lines for each stem. Ages of Students Key: 1|8 = 18 1 1 2 2 3 3 4 4 5 5 8 8 8 9 9 9 0 0 1 1 1 2 4 7 9 9 0 0 2 2 3 4 From this graph, we can conclude that more than 50% of the data lie between 20 and 34. 7 8 9 4 6 9 1 4

Dot Plot In a dot plot, each data entry is plotted, using a point, above a horizontal axis. Example: Use a dot plot to display the ages of the 30 students in the statistics class. Ages of Students Continued.

15 18 21 24 27 30 33 36 39 42 45 48 51 57 54 Dot Plot Ages of Students From this graph, we can conclude that most of the values lie between 18 and 32.

Pie Chart A piechart is a circle that is divided into sectors that represent categories. The area of each sector is proportional to the frequency of each category. Accidental Deaths in the USA in 2002 (Source: US Dept. of Transportation) Continued.

Pie Chart To create a pie chart for the data, find the relative frequency (percent) of each category. n = 75,200 Continued.

Pie Chart Next, find the central angle. To find the central angle, multiply the relative frequency by 360°. Continued.

Pie Chart Firearms 1.9% Ingestion 3.9% Fire 5.6% Drowning 6.1% Poison 8.5% Motor vehicles 57.8% Falls 16.2%

Pareto Chart A Pareto chart is a vertical bar graph is which the height of each bar represents the frequency. The bars are placed in order of decreasing height, with the tallest bar to the left. Accidental Deaths in the USA in 2002 (Source: US Dept. of Transportation) Continued.

Ingestion of Food/Object Pareto Chart Accidental Deaths 45000 40000 35000 30000 25000 20000 15000 10000 Poison 5000 MotorVehicles Firearms Falls Poison Drowning Fire

Scatter Plot When each entry in one data set corresponds to an entry in another data set, the sets are called paired data sets. In a scatter plot, the ordered pairs are graphed as points in a coordinate plane. The scatter plot is used to show the relationship between two quantitative variables. The following scatter plot represents the relationship between the number of absences from a class during the semester and the final grade. Continued.

Absences Grade x 8 2 5 12 15 9 6 y 78 92 90 58 43 74 81 Final grade (y) 100 90 80 70 60 50 40 Absences (x) 16 0 2 4 6 8 10 12 14 Scatter Plot From the scatter plot, you can see that as the number of absences increases, the final grade tends to decrease.

Times Series Chart A data set that is composed of quantitative data entries taken at regular intervals over a period of time is a timeseries. A timeseries chart is used to graph a time series. Example: The following table lists the number of minutes Robert used on his cell phone for the last six months. Construct a time series chart for the number of minutes used. Continued.

Robert’s Cell Phone Usage 250 200 150 Minutes 100 50 0 Jan Feb Mar Apr May June Month Times Series Chart

Population mean: Sample mean: “mu” “x-bar” Mean A measure of central tendency is a value that represents a typical, or central, entry of a data set. The three most commonly used measures of central tendency are the mean, the median, and the mode. The mean of a data set is the sum of the data entries divided by the number of entries.

Mean • Example: • The following are the ages of all seven employees of a small company: 53 32 61 57 39 44 57 Calculatethe population mean. Add the ages and divide by 7. The mean age of the employees is 49 years.

Median The median of a data set is the value that lies in the middle of the data when the data set is ordered. If the data set has an odd number of entries, the median is the middle data entry. If the data set has an even number of entries, the median is the mean of the two middle data entries. • Example: • Calculatethe median age of the seven employees. 53 32 61 57 39 44 57 To find the median, sort the data. 32 39 44 53 57 57 61 The median age of the employees is 53 years.

Mode The mode of a data set is the data entry that occurs with the greatest frequency. If no entry is repeated, the data set has no mode. If two entries occur with the same greatest frequency, each entry is a mode and the data set is called bimodal. • Example: • Findthe mode of the ages of the seven employees. 53 32 61 57 39 44 57 The mode is 57 because it occurs the most times. An outlier is a data entry that is far removed from the other entries in the data set.

Comparing the Mean, Median and Mode • Example: • A 29-year-old employee joins the company and the ages of the employees are now: 53 32 61 57 39 44 5729 Recalculate the mean, the median, and the mode. Which measure of central tendency was affected when this new age was added? Mean = 46.5 The mean takes every value into account, but is affected by the outlier. Median = 48.5 The median and mode are not influenced by extreme values. Mode = 57

Weighted Mean A weighted mean is the mean of a data set whose entries have varying weights. A weighted mean is given by where w is the weight of each entry x. Example: Grades in a statistics class are weighted as follows: Tests are worth 50% of the grade, homework is worth 30% of the grade and the final is worth 20% of the grade. A student receives a total of 80 points on tests, 100 points on homework, and 85 points on his final. What is his current grade? Continued.

Weighted Mean Begin by organizing the data in a table. The student’s current grade is 87%.

The mean of a frequency distribution for a sample is approximated by where x and f are the midpoints and frequencies of the classes. Mean of a Frequency Distribution Example: The following frequency distribution represents the ages of 30 students in a statistics class. Find the mean of the frequency distribution. Continued.

Class midpoint Mean of a Frequency Distribution The mean age of the students is 30.3 years.

Shapes of Distributions A frequency distribution is symmetric when a vertical line can be drawn through the middle of a graph of the distribution and the resulting halves are approximately the mirror images. A frequency distribution is uniform (or rectangular) when all entries, or classes, in the distribution have equal frequencies. A uniform distribution is also symmetric. A frequency distribution is skewed if the “tail” of the graph elongates more to one side than to the other. A distribution is skewedleft (negativelyskewed) if its tail extends to the left. A distribution is skewedright (positivelyskewed) if its tail extends to the right.

Symmetric Distribution Income 5 4 f 3 2 1 0 $25000 mean = median = mode = $25,000 10 Annual Incomes

Skewed Left Distribution 5 Income 4 f 3 2 1 0 $25000 10 Annual Incomes mean = $23,500 median = mode = $25,000 Mean < Median

Skewed Right Distribution 5 Income 4 f 3 2 1 0 $25000 10 Annual Incomes mean = $121,500 median = mode = $25,000 Mean > Median