# Descriptive Statistics. Frequency Distributions and Their Graphs 2.1. Frequency Distributions. Chapter 2

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1 Chapter Descriptive Statistics.1 Frequency Distributions and Their Graphs 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. Lower Upper Class Limits Class Frequency, f 5 Frequencies Larson & Farber, Elementary Statistics: Picturing the World, e 1

2 Frequency Distributions The class width is the distance between lower (or upper) limits of consecutive classes. 5 1 = 9 5 = 1 9 = 17 1 = Class Frequency, f The class width is. The range is the difference between the maximum and minimum data entries. Larson & Farber, Elementary Statistics: Picturing the World, e Constructing a Frequency Distribution Guidelines 1. Decide on the number of classes to include. The number of classes should be between 5 and 0; 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. 5. Count the tally marks to find the total frequency f for each class. Larson & Farber, Elementary Statistics: Picturing the World, e 5 Constructing a Frequency Distribution The following data represents the ages of 0 students in a statistics class. Construct a frequency distribution that has five classes Larson & Farber, Elementary Statistics: Picturing the World, e 6

3 Constructing a Frequency Distribution Example continued: 1. The number of classes (5) is stated in the problem.. The minimum data entry is 18 and maximum entry is 5, so the range is 6. Divide the range by the number of classes to find the class width. Class width = 6 5 = 7. Round up to 8. Larson & Farber, Elementary Statistics: Picturing the World, e 7 Constructing a Frequency Distribution Example continued:. 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, 6,,, and 50. The upper class limits are 5,, 1, 9, and 57.. 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. Larson & Farber, Elementary Statistics: Picturing the World, e 8 Constructing a Frequency Distribution Example continued: Ages Class Tally Frequency, f f = 0 Number of students Check that the sum equals the number in the sample. Larson & Farber, Elementary Statistics: Picturing the World, e 9

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 = (Lower class limit) + (Upper class limit) Class 1 Frequency, f Midpoint.5 Midpoint = 1+ 5 = =.5 Larson & Farber, Elementary Statistics: Picturing the World, e 10 Midpoint Find the midpoints for the frequency distribution. Class Frequency, f 1 8 f = 0 Midpoint = = 1.5 Larson & Farber, Elementary Statistics: Picturing the World, e 11 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 = Class frequency Sample size f = n Class 1 Frequency, f f = 18 Relative frequency f = n Relative Frequency 0. = Larson & Farber, Elementary Statistics: Picturing the World, e 1

5 Relative Frequency Find the relative frequencies for the frequency distribution. Class Frequency, f f = 0 Relative Frequency f = 1 n Portion of students f n 1 = 0 0. Larson & Farber, Elementary Statistics: Picturing the World, e 1 Cumulative Frequency The cumulative frequency of a class is the sum of the frequency for that class and all the previous classes. Class Frequency, f f = 0 Cumulative Frequency Total number students of Larson & Farber, Elementary Statistics: Picturing the World, e 1 Frequency Histogram 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.. 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. Larson & Farber, Elementary Statistics: Picturing the World, e 15 5

6 Class Boundaries Find the class boundaries for the frequency distribution. The distance from the upper limit of the first class to the lower limit of the second class is 1. Half this distance is 0.5. Class Frequency, f f = 0 Class Boundaries Larson & Farber, Elementary Statistics: Picturing the World, e 16 Frequency Histogram Draw a frequency histogram for the frequency distribution. Use the class boundaries. f Broken axis Age (in years) Larson & Farber, Elementary Statistics: Picturing the World, e 17 Frequency Polygon A frequency polygon is a line graph that emphasizes the continuous change in frequencies. f Broken axis Age (in years) Line is extended to the x-axis Midpoints Larson & Farber, Elementary Statistics: Picturing the World, e 18 6

7 Relative Frequency Histogram A relative frequency histogram has the same shape and the same horizontal scale as the corresponding frequency histogram. Relative frequency (portion of students) Age (in years) Larson & Farber, Elementary Statistics: Picturing the World, e 19 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. Cumulative frequency (portion of students) Age (in years) The graph ends at the upper boundary of the last class. Larson & Farber, Elementary Statistics: Picturing the World, e 0. More Graphs and Displays 7

8 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. The following data represents the ages of 0 students in a statistics class. Display the data in a stem-and-leaf plot Larson & Farber, Elementary Statistics: Picturing the World, e Stem-and-Leaf Plot Key: 1 8 = 18 Most of the values lie between 0 and 9. This graph allows us to see the shape of the data as well as the actual values. Larson & Farber, Elementary Statistics: Picturing the World, e Stem-and-Leaf Plot Construct a stem-and-leaf plot that has two lines for each stem Key: 1 8 = 18 From this graph, we can conclude that more than 50% of the data lie between 0 and. Larson & Farber, Elementary Statistics: Picturing the World, e 8

9 Dot Plot In a dot plot, each data entry is plotted, using a point, above a horizontal axis. Use a dot plot to display the ages of the 0 students in the statistics class Larson & Farber, Elementary Statistics: Picturing the World, e Dot Plot From this graph, we can conclude that most of the values lie between 18 and. Larson & Farber, Elementary Statistics: Picturing the World, e 6 Pie Chart A pie chart is a circle that is divided into sectors that represent categories. The area of each sector is proportional to the frequency of each category. (Source: US Dept. of Transportation) Accidental Deaths in the USA in 00 Type Frequency Motor Vehicle Falls Poison Drowning Fire Ingestion of Food/Object Firearms,500 1,00 6,00,600,00,900 1,00 Larson & Farber, Elementary Statistics: Picturing the World, e 7 9

10 Pie Chart To create a pie chart for the data, find the relative frequency (percent) of each category. Motor Vehicle Falls Poison Drowning Fire Type Ingestion of Food/Object Firearms Frequency,500 1,00 6,00,600,00,900 1,00 n = 75,00 Relative Frequency Larson & Farber, Elementary Statistics: Picturing the World, e 8 Pie Chart Next, find the central angle. To find the central angle, multiply the relative frequency by 60. Type Frequency Relative Frequency Angle Motor Vehicle, Falls 1, Poison 6, Drowning, Fire, Ingestion of Food/Object, Firearms 1, Larson & Farber, Elementary Statistics: Picturing the World, e 9 Pie Chart Fire 5.6% Ingestion.9% Firearms 1.9% Drowning 6.1% Poison 8.5% Falls 16.% Motor vehicles 57.8% Larson & Farber, Elementary Statistics: Picturing the World, e 0 10

11 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. (Source: US Dept. of Transportation) Accidental Deaths in the USA in 00 Type Frequency Motor Vehicle Falls Poison Drowning Fire Ingestion of Food/Object Firearms,500 1,00 6,00,600,00,900 1,00 Larson & Farber, Elementary Statistics: Picturing the World, e 1 Pareto Chart Accidental Deaths Poison Motor Vehicles Falls Poison Drowning Fire Ingestion of Food/Object Firearms Larson & Farber, Elementary Statistics: Picturing the World, e 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. Larson & Farber, Elementary Statistics: Picturing the World, e 11

12 Scatter Plot Final grade (y) Absences (x) Absences x Grade y From the scatter plot, you can see that as the number of absences increases, the final grade tends to decrease. Larson & Farber, Elementary Statistics: Picturing the World, e Times Series Chart A data set that is composed of quantitative data entries taken at regular intervals over a period of time is a time series. A time series chart is used to graph a time series. 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. Month January February March April May June Minutes Larson & Farber, Elementary Statistics: Picturing the World, e 5 Times Series Chart 50 Robert s Cell Phone Usage 00 Minutes Jan Feb Mar Apr May June Month Larson & Farber, Elementary Statistics: Picturing the World, e 6 1

13 . Measures of Central Tendency 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. Population mean: x µ = Sample mean: N x x = n mu x-bar Larson & Farber, Elementary Statistics: Picturing the World, e 8 Calculate the population mean. Mean The following are the ages of all seven employees of a small company: x µ = N = 7 = 9 year s Add the ages and divide by 7. The mean age of the employees is 9 years. Larson & Farber, Elementary Statistics: Picturing the World, e 9 1

14 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. Calculate the median age of the seven employees To find the median, sort the data The median age of the employees is 5 years. Larson & Farber, Elementary Statistics: Picturing the World, e 0 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. Find the mode of the ages of the seven employees 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. Larson & Farber, Elementary Statistics: Picturing the World, e 1 Comparing the Mean, Median and Mode A 9-year-old employee joins the company and the ages of the employees are now: Recalculate the mean, the median, and the mode. Which measure of central tendency was affected when this new age was added? Mean = 6.5 Median = 8.5 Mode = 57 The mean takes every value into account, but is affected by the outlier. The median and mode are not influenced by extreme values. Larson & Farber, Elementary Statistics: Picturing the World, e 1

15 Weighted Mean A weighted mean is the mean of a data set whose entries have varying weights. A weighted mean is given by ( x w ) x = w where w is the weight of each entry x. Grades in a statistics class are weighted as follows: Tests are worth 50% of the grade, homework is worth 0% of the grade and the final is worth 0% 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? Larson & Farber, Elementary Statistics: Picturing the World, e Weighted Mean Begin by organizing the data in a table. Source Tests Homework Final Score, x Weight, w xw ( x w ) x = w 87 = = The student s current grade is 87%. Larson & Farber, Elementary Statistics: Picturing the World, e Mean of a Frequency Distribution The mean of a frequency distribution for a sample is approximated by ( x f ) x = Note that n = f n where x and f are the midpoints and frequencies of the classes. The following frequency distribution represents the ages of 0 students in a statistics class. Find the mean of the frequency distribution. Larson & Farber, Elementary Statistics: Picturing the World, e 5 15

16 Mean of a Frequency Distribution Class midpoint Class x f 1 8 n = 0 (x f ) Σ = ( x f ) 909 x = = = 0. n 0 The mean age of the students is 0. years. Larson & Farber, Elementary Statistics: Picturing the World, e 6 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 skewed left (negatively skewed) if its tail extends to the left. A distribution is skewed right (positively skewed) if its tail extends to the right. Larson & Farber, Elementary Statistics: Picturing the World, e 7 Symmetric Distribution 10 Annual Incomes 15,000 0,000,000,000 5,000 5,000 6,000 8,000 0,000 5,000 mean = median = mode = \$5,000 f Income \$5000 Larson & Farber, Elementary Statistics: Picturing the World, e 8 16

17 Skewed Left Distribution 10 Annual Incomes 0 0,000,000,000 5,000 5,000 6,000 8,000 0,000 5,000 f Income \$5000 mean = \$,500 median = mode = \$5,000 Mean < Median Larson & Farber, Elementary Statistics: Picturing the World, e 9 Skewed Right Distribution 10 Annual Incomes 15,000 0,000,000,000 5,000 5,000 6,000 8,000 0,000 1,000,000 mean = \$11,500 median = mode = \$5,000 f Income \$5000 Mean > Median Larson & Farber, Elementary Statistics: Picturing the World, e 50 Summary of Shapes of Distributions Symmetric Uniform Mean = Median Skewed right Skewed left Mean > Median Mean < Median Larson & Farber, Elementary Statistics: Picturing the World, e 51 17

18 . Measures of Variation Range The range of a data set is the difference between the maximum and minimum date entries in the set. Range = (Maximum data entry) (Minimum data entry) The following data are the closing prices for a certain stock on ten successive Fridays. Find the range. Stock The range is = 11. Larson & Farber, Elementary Statistics: Picturing the World, e 5 Deviation The deviation of an entry x in a population data set is the difference between the entry and the mean µ of the data set. Deviation of x = x µ The following data are the closing prices for a certain stock on five successive Fridays. Find the deviation of each price. The mean stock price is µ = 05/5 = 61. Stock x Σx = 05 Deviation x µ = = = = = 6 Σ(x µ) = 0 Larson & Farber, Elementary Statistics: Picturing the World, e 5 18

19 Variance and Standard Deviation The population variance of a population data set of N entries is ( x µ ) Population variance = σ =. N sigma squared The population standard deviation of a population data set of N entries is the square root of the population variance. ( x µ ) Population standard deviation = σ = σ =. N sigma Larson & Farber, Elementary Statistics: Picturing the World, e 55 Finding the Population Standard Deviation Guidelines In Words 1. Find the mean of the population data set.. Find the deviation of each entry.. Square each deviation.. Add to get the sum of squares. 5. Divide by N to get the population variance. 6. Find the square root of the variance to get the population standard deviation. In Symbols x µ = N x µ ( x µ ) SS ( ) x = x µ ( ) x µ σ = N σ = ( x µ ) N Larson & Farber, Elementary Statistics: Picturing the World, e 56 Finding the Sample Standard Deviation Guidelines In Words 1. Find the mean of the sample data set.. Find the deviation of each entry.. Square each deviation.. Add to get the sum of squares. 5. Divide by n 1 to get the sample variance. 6. Find the square root of the variance to get the sample standard deviation. In Symbols x x = n x x ( x x) S S ( ) x = x x ( ) x x s = n 1 s = ( x x) n 1 Larson & Farber, Elementary Statistics: Picturing the World, e 57 19

20 Finding the Population Standard Deviation The following data are the closing prices for a certain stock on five successive Fridays. The population mean is 61. Find the population standard deviation. Always positive! Stock x Σx = 05 Deviation x µ Σ(x µ) = 0 Squared (x µ) Σ(x µ) = 7 SS = Σ(x µ) = 7 ( ) x µ 7 σ = = = 1.8 N 5 ( x µ ) σ = = N σ \$.85 Larson & Farber, Elementary Statistics: Picturing the World, e 58 Interpreting Standard Deviation When interpreting standard deviation, remember that is a measure of the typical amount an entry deviates from the mean. The more the entries are spread out, the greater the standard deviation. Frequency x = s = Data value Frequency x = s = 0 6 Data value Larson & Farber, Elementary Statistics: Picturing the World, e 59 Empirical Rule ( %) Empirical Rule For data with a (symmetric) bell-shaped distribution, the standard deviation has the following characteristics. 1. About 68% of the data lie within one standard deviation of the mean.. About 95% of the data lie within two standard deviations of the mean.. About 99.7% of the data lie within three standard deviation of the mean. Larson & Farber, Elementary Statistics: Picturing the World, e 60 0

21 Empirical Rule ( %) 99.7% within standard deviations 95% within standard deviations 68% within 1 standard deviation % %.5%.5% 1.5% 1.5% Larson & Farber, Elementary Statistics: Picturing the World, e 61 Using the Empirical Rule The mean value of homes on a street is \$15 thousand with a standard deviation of \$5 thousand. The data set has a bell shaped distribution. Estimate the percent of homes between \$10 and \$10 thousand. 68% µ σ µ µ + σ 68% of the houses have a value between \$10 and \$10 thousand. Larson & Farber, Elementary Statistics: Picturing the World, e 6 Chebychev s Theorem The Empirical Rule is only used for symmetric distributions. Chebychev s Theorem can be used for any distribution, regardless of the shape. Larson & Farber, Elementary Statistics: Picturing the World, e 6 1

22 Chebychev s Theorem The portion of any data set lying within k standard deviations (k > 1) of the mean is at least 1 1. k For k = : In any data set, at least 1 1 = 1 1 =, or 75%, of the data lie within standard deviations of the mean. For k = : In any data set, at least 1 1 = 1 1 = 8, or 88.9%, of the 9 9 data lie within standard deviations of the mean. Larson & Farber, Elementary Statistics: Picturing the World, e 6 Using Chebychev s Theorem The mean time in a women s 00-meter dash is 5. seconds with a standard deviation of. sec. At least 75% of the women s times will fall between what two values? standard deviations µ At least 75% of the women s 00-meter dash times will fall between 8 and 56.8 seconds. Larson & Farber, Elementary Statistics: Picturing the World, e 65 Standard Deviation for Grouped Data Sample standard deviation = s = where n = Σf is the number of entries in the data set, and x is the data value or the midpoint of an interval. ( x x ) f n 1 The following frequency distribution represents the ages of 0 students in a statistics class. The mean age of the students is 0. years. Find the standard deviation of the frequency distribution. Larson & Farber, Elementary Statistics: Picturing the World, e 66

23 Standard Deviation for Grouped Data The mean age of the students is 0. years. Class x f n = 0 x x (x x ) (x x ) f = s = ( x x ) f n = = = The standard deviation of the ages is 10. years. Larson & Farber, Elementary Statistics: Picturing the World, e 67.5 Measures of Position Quartiles The three quartiles, Q 1, Q, and Q, approximately divide an ordered data set into four equal parts. Median Q 1 Q Q Q 1 is the median of the data below Q. Q is the median of the data above Q. Larson & Farber, Elementary Statistics: Picturing the World, e 69

24 Finding Quartiles The quiz scores for 15 students is listed below. Find the first, second and third quartiles of the scores Order the data. Lower half Upper half Q 1 Q Q About one fourth of the students scores 7 or less; about one half score or less; and about three fourths score 8 or less. Larson & Farber, Elementary Statistics: Picturing the World, e 70 Interquartile Range The interquartile range (IQR) of a data set is the difference between the third and first quartiles. Interquartile range (IQR) = Q Q 1. The quartiles for 15 quiz scores are listed below. Find the interquartile range. Q 1 = 7 Q = Q = 8 (IQR) = Q Q 1 = 8 7 = 11 The quiz scores in the middle portion of the data set vary by at most 11 points. Larson & Farber, Elementary Statistics: Picturing the World, e 71 Box and Whisker Plot A box-and-whisker plot is an exploratory data analysis tool that highlights the important features of a data set. The five-number summary is used to draw the graph. The minimum entry Q 1 Q (median) Q The maximum entry Use the data from the 15 quiz scores to draw a box-and-whisker plot Larson & Farber, Elementary Statistics: Picturing the World, e 7

25 Box and Whisker Plot Five-number summary The minimum entry Q 1 Q (median) Q The maximum entry Quiz Scores Larson & Farber, Elementary Statistics: Picturing the World, e 7 Percentiles and Deciles Fractiles are numbers that partition, or divide, an ordered data set. Percentiles divide an ordered data set into 100 parts. There are 99 percentiles: P 1, P, P P 99. Deciles divide an ordered data set into 10 parts. There are 9 deciles: D 1, D, D D 9. A test score at the 80th percentile (P 80 ), indicates that the test score is greater than 80% of all other test scores and less than or equal to 0% of the scores. Larson & Farber, Elementary Statistics: Picturing the World, e 7 Standard Scores The standard score or z-score, represents the number of standard deviations that a data value, x, falls from the mean, µ. value mean x µ z = = standard deviation σ The test scores for all statistics finals at Union College have a mean of 78 and standard deviation of 7. Find the z-score for a.) a test score of 85, b.) a test score of 70, c.) a test score of 78. Larson & Farber, Elementary Statistics: Picturing the World, e 75 5

26 Example continued: a.) µ = 78, σ = 7, x = 85 Standard Scores x µ z = σ = = 1.0 This score is 1 standard deviation higher 7 than the mean. b.) µ = 78, σ = 7, x = 70 x µ z = σ = = 1.1 This score is 1.1 standard deviations 7 lower than the mean. c.) µ = 78, σ = 7, x = 78 x µ z = σ = = 0 This score is the same as the mean. 7 Larson & Farber, Elementary Statistics: Picturing the World, e 76 Relative Z-Scores John received a 75 on a test whose class mean was 7. with a standard deviation of.5. Samantha received a 68.6 on a test whose class mean was 65 with a standard deviation of.9. Which student had the better test score? John s z-score x µ z = = σ.5 = 0. Samantha s z-score x µ z = = σ.9 = 0.9 John s score was 0. standard deviations higher than the mean, while Samantha s score was 0.9 standard deviations higher than the mean. Samantha s test score was better than John s. Larson & Farber, Elementary Statistics: Picturing the World, e 77 6

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