Graphical Descriptions of Data and Variables

DATA SET: Collection of data values or data points    N = number of data points or values in the set FREQUENCY: the number of times a specific data point/ value is repeated RELATIVE FREQUENCY: gives frequency for each data value as a percentage of the total data set VARIABLE: any characteristic that varies with the members of a population 2 BASIC CLASSIFICATIONS OF VARIABLES: NUMERICAL (Quantitative) Variable: variable that is a measureable quantity 1. CONTINUOUS: difference between values of a numerical variable are arbitrarily small o Examples: Height, Foot Size, Mile Run Time 2. DISCRETE: values of numerical variable change by minimum increments o Examples Shoe Size, IQ, SAT Score, Points scored in a basketball game CATEGORICAL (Qualitative) Variables: variable that cannot be measured numerically  Examples: Race, Nationality, Gender, Hair Color Identify the following variables as categorical, discrete or continuous. 1) Occupation = 3) Region of residence = 5) Education level = Categorical Categorical Categorical 6) Number of automobiles owned 2) Weight = 4) Family Size = Discrete = Discrete Continuous How can we represent a data set? 1) LIST: All N data points or values are listed (can be ascending, descending, or random orders) 2) FREQUENCY TABLE: data values paired with the number of times that value is repeated. o Do not list data values of frequency zero 3) A. BAR GRAPH: Plots the data values, in increasing order, and frequency for each data point. o Axes = Data Values and Frequencies (Usually Frequencies are vertical) o Bars DO NOT TOUCH. o Bar Graph is a more visual representation of a frequency table and shows 0 frequencies. B. HISTOGRAMS: A type of bar graph for continuous numerical variables, in which the primary difference is that the bars will now touch each other or can use relative frequencies. The dimensions of the bars may change to include different data values and combined frequencies. CLASS INTERVAL: Grouping together data points into categories (score or data ranges)  Endpoint Convention: If a data value falls where two bars meet, which bar does that data value belong with. 4) PIE CHART: Uses relative frequencies (percentages) for the sectors of each data group 14.1 and 14.2 PRACTICE PROBLEMS Student ID 1362 1486 1721 1932 2489 2766 2877 2964 3217 3588 3780 3921 Score 90 90 70 60 60 80 80 60 80 100 90 90 Student ID 4315 4719 4951 5321 5872 6533 6921 8317 8854 8964 9158 9347 1) The table below contains the scores on a Chemistry 103 final exam consisting of ten questions worth ten points each. Complete the frequency table for this exam. Score 10 0 70 40 50 70 70 90 80 70 90 80 0=1 10 = 1 40 = 1 50 = 1 60 = 3 70 = 5 80 = 5 90 = 6 100 = 1 2) Suppose the grading scale for the Chemistry exam is A: 80-100, B: 70-79, C: 60-69, D: 50-59 and F: 0-49. Find the grade distribution for the exam. 3) The table to the right shows the grade distribution for a recent civics test. Find the relative frequency for each grade from the civics test. Frequency Table of Chemistry Grade Distribution Grade A B C D F Frequency 5 3 1 3 Frequency Table for Civics Grade Distribution Grade A B C D F Frequency 3 7 11 2 1 Relative 4) The bar graph describes the scores of a group of students on a 10-point math quiz. 4a. How many students took the math quiz? 40 4b. What percentage of the students scored 2 points? 0% 4c. If a grade of 6 or more was needed to pass the quiz, what percentage of the students passed? 30/40  75% 12 12.5 29.16 45.83 8.33 4.16 5) The pie chart to the right shows the possible causes of death among 18-22 year olds. 5a. Is cause of death a quantitative or qualitative variable? 5b. Based on the data provided in the pie chart, estimate the number of 18- to 22-year-olds in the population studied who died in as the result of an accident (round to the nearest whole number). 0.44191(19,548) = 8,638.45668 or 8,638 6) The pie chart to the right represents the breakdown of a federal government’s $2.9 trillion budget in the last fiscal year. Calculate the size of the central angle in degrees for each wedge of the pie chart (round to the nearest tenth). 83.160 = .231*3600 77.760 = .216*3600 60.640 = .169*3600 31.320 = .087*3600 106.920 = .297*3600 7) The following is the frequency table for the musical aptitude scores for 1st grade students. 0 = no musical aptitude Aptitude Score 0 1 2 3 4 5 5 = extremely talented Frequency 24 16 20 12 5 3 7a. What are the data values in this problem? Aptitude SCORES 7b. How many students took the aptitude test? 24 + 16 + 20 + 12 + 5 + 3 = 80 7c. What percent of the students tested showed no musical aptitude? 24/80*100 = 30% 7d. What percent of students showed approximately average musical aptitude (Scored a 2 or 3)? (20 + 12)/80 * 100 = 40% 14.3 Numerical Summaries of Data Data points/Values Notations Xi – The upper case X with the subscript i represents the ith data point in a population data set. xi – The lower case x with the subscript i represents the ith data point in a sample data set. i – The subscript letter i is used to locate (or “indicate”) its position in a set of data that is sorted from the least value to the greatest value. Number of data points N – The upper case N is used to represent the number of data points in a population data set n – The lower case n is used to represent the number of data points in a sample data set. MEAN: The mean (or average) of a data set is found by dividing the sum of all values in the data set by the number of values in the data set • Data does not have to be sorted to find the mean μ – The lower case Greek letter μ is used to represent the mean (or average) of a population data set. x – The lower case x with a bar over the top (read “x-bar”) is used to represent the mean of a sample data set. MEDIAN: If we sort the data in order from least to greatest, the median is the data point that is found in the exact middle of the sorted data. • Data MUST be sorted to find the median M – The upper case M is used to represent the median of any data set. Finding the median: “Count inward from the min and max until you end up in the middle” or divide If the number of data points n is ODD, then it is an actual data value. ( Xn/2↑) If the number of data points n is EVEN, then it is the average of the two middle data values( [Xn/2 + Xn/2 + 1]/2) MODE: The mode of a data set is the value that has the highest frequency of occurrence (repeated). • • There can be multiple modes in a data set if two (or more) data points have the highest frequency. If a data set has no repeated values, then there is no mode for that data set. 1. Consider the sample data set {–7.8, –4.5, –14.8, 5.8, 5.8, 0.2, –14.8, –6.6}. a. What is the size of the data set? 8 b. Sort the data set from least to greatest. __-14.8, -14.8, -7.8, -6.6, -4.5, 0.2, 5.8, 5.8________ c. What is the value of the first data point? d. What is the value of the fifth data point? x1 = __-14.8_______ x5 = ___-4.5______ e. Find the mean. f. Find the median. g. Find the mode. x = ____-4.5875_______ M = ____-5.55______ mode = ____-5.8, -14.8_____ 2. Find the mean, median and mode of each sample data set. a. { 3, 4, 5, 6, 7, 8, 9, 10} x = __________ M = __6.5_____ mode = __None_____ b. {3, 5, 8, 11, 14, 15, 16, 17, 18} x = __________ M = __14_____ mode = __None____ 3. The frequency table to the shows the scores of quiz consisting of three questions worth 10 points each. a. What is the size of the data set? n = __40____ b. Find the mean, median and mode of the data set. x = __15.5____ M = ___15____ mode = ___10__ Quiz Score Frequency 0 6 10 14 20 12 30 8 PERCENTILE: the pth percentile of a data set is a data value such that p% of the data is at or below that value and the rest of data is at or above it. FINDING PERCENTILE: There are three steps to finding the pth percentile. Step 1. SORT the data xi in order from the least value to the greatest value. Step 2. Find the locator i for the pth percentile. (Location based on total number of values) p Locator : i = ×N 100 Step 3. Find the pth percentile. The percentile depends on whether or not the locator i is a whole number.  If i is a whole number, then the pth percentile is the average of the ith data value, Xi, and the data X + X i +1 value after it (i+1st data value), Xi+1: percentile = i 2  If i is NOT a whole number, we round up i to the next whole number, i+ and the pth percentile is Xi+. Percentile = Next Available data value after i Find the 40th percentile and 75th percentile for {1, 2, 3, 4, 5, 0, 2, 3, 6, 8} Step #1: {0, 1, 2, 2, 3, 3, 4, 5, 6, 8} 40th Percentile Locator: i = (40/100)*10 = 4 Step #2: 75th Percentile Locator: i = (75/100)*10 = 7.5 If i is a whole number, 40th Percentile: I = 4  average X4 and X5 = (2 +3)/ 2 = 2.5 Step #3: If i is not a whole number, 75th Percentile: i = 7.5  i+ = 8 and X8 = 5 Practice Problems: Consider the sorted GPAs 3.33 3.35 3.41 3.42 3.45 3.57 3.62 3.65 3.67 3.71 3.76 3.82 3.88 3.91 1a) Find the 80th percentile. (.8)*15 = 12 d12.5 = (3.82 + 3.88)/2 = 3.85 4.0 1b)Find the 55th percentile. (.55)*15 = 8.25; d9 = 3.67 #2) Athletes with GPAs in the 80th percentile or above will earn a $5000 scholarship. Which GPAs earned a $5000 scholarship? 3.85 or above (3.88, 3.91 and 4.00) each won the $5000 scholarship. #3) Athletes with GPAs from the 55th to the 80th percentile will get a $2000 scholarship. Which GPAs earned a $2000 scholarship? at least 3.67 and less than 3.85 (3.67, 3.71, 3.76 and 3.82) QUARTILE: units of 25% of the data values  1st Quartile = 25th Percentile or Q1  “Halfway between Median and Minimum”  2nd Quartile = 50th Percentile or MEDIAN, M  3rd Quartile = 75th Percentile or Q3  “Halfway between Median and Maximum” 1) 2) 3) 4) 5) FIVE-NUMBER SUMMARY MIN: Minimum Value (0 Percentile) Q1: 1st Quartile (25th Percentile) M or Q2: Median (50th Percentile) Q3: 3rd Quartile (75th Percentile) MAX or Q4: Maximum Value (100th Percentile) VISUALIZING FIVE-NUMBER SUMMARY - BOX PLOT (Box and Whisker Plot) Boxes = Range: Q1 to Median and Median to Q3 MIN Q1 Whiskers = Min to Q1 and Q3 to Max Q3 M MAX Getting Five Number Summary using Calculator: One-Variable Statistics [STAT]  [EDIT]  L1 Clear DO NOT Delete Lists: Highlight L1  [Clear], then input data values [STAT]  [CALC]  [1: 1-Var STATS]  [ENTER] • • • • • • • • • x represents mean Hint: Use L1, L2 for frequency tables before Enter Sx represents standard deviation of sample (divides by n - 1) σx represents standard deviation of population (divides by n) ** USE THIS ONE** n = number of entries minX = Minimum Data Value Q1 = 1st Quartile med = median Q3 = 3rd Quartile maxX = Maximum Data Value Find the five-number summary, mean, and mode of the data: 1) {65, 68, 70, 71, 73, 73, 74, 76, 78, 81, 81, 85, 86, 87, 89, 90, 91, 91, 93, 95} Min = 65 Q1 = 73 M = 81 Q3 = 89.5 MEAN = 80.85 Max = 95 Mode = 73, 81, 91 2) Frequency Table for Chemistry 103 Exam Score Frequency Min = 10 10 50 1 3 Q1 = 60 Mean = 66.25 60 70 7 7 M = 70 80 4 100 2 Q3 = 75 Mode = 60, 70 Max = 100 RANGE, R:  14.4 Measures of Spread R = Max – Min Represents the spread of ALL data values IQR = Q3 - Q1 INTERQUARTILE RANGE, IQR:  Represents the spread of the MIDDLE 50% of the data values OUTLIERS: an extreme data point that does not fit into the overall pattern CALCULATING AN OUTLIER: Use the IQR Value > Q3 + 1.5 IQR or Value < Q1 - 1.5 IQR Use information from the five number summary to calculate. Example #1: For {-7, -5, -4, -2, 0, 1, 3, 4, 5, 6, 7, 8, 8, 9}, Identify the following 1a. Five Number Summary: Min = -7, Q1 = -2, M =3.5, Q3 = 7, Max = 9 1b. Mean: 2.36 1c. Mode: 8 1d. Range: 9 - -7 = 16 1e. IQR: 7 - -2 = 9 1f.Upper Outlier Values > 7 + 1.5(9) = 20.5 1g. Lower Outlier Values < -2 – 1.5(9) = -15.5 STANDARD DEVIATION: The most important and most commonly used measure of spread. In simple terms, the standard deviation of a data set is the “average deviation from the mean.” σx – The lower case Greek letter σ is used to represent the standard deviation of a population data set. Sx – The lower case s is used to represent the standard deviation of a sample data set. Calculating standard deviation is a multi-step process. There is a difference between the calculations for population and sample standard deviations. Step 1 – Find the MEAN of the data set. Step 2 – Find the DEVIATION (difference) from the mean of each value in the data set. Deviation = (Data Value – Mean) Step 3 – Find the VARIANCE of the data set. Square the deviations and add them together, then divide that total by N = size of population or n – 1 = one less than sample size. N **Population variance**: σ2 = ∑ ( X i − µ)2 i =1 n sample variance: s2 = ∑(x i =1 i − x)2 N n −1 **VARIANCE can be found by SQUARING the STANDARD DEVIATION.** Step 5 – Find the STANDARD DEVIATION the data set. N **SD of population**: σX = ∑(X i =1 i − µ )2 n ∑(x i − x)2 SD of sample: S x = i =1 n−1 N Note: difference in divisors between the population and sample standard deviations. The FIVE-NUMBER SUMMARY gives both of the standard deviations, σx and Sx CHAPTER 14: DESCRIPTIVE STATISTICS GENERAL PRACTICE PROBLEMS – USE CALCULATOR For problems #1 - #2, find the (a) mean, (b) standard deviation, and (c) five-number summary. 1) {12, 18, 19, 23, 27, 31, 36} 2) {41.5, 44.2, 51.9, 58.4, 63.7, 68.1} (a) Mean ≈ 23.714 (b) Standard Deviation ≈7.63 (a) Mean ≈ 54.63 (b) Standard Deviation ≈9.717 (c) Min = 12, Q1 = 18, Med = 23, Q3 = 31 Max = 36 (c) Min = 41.5, 44.2, Med = 55.15, 63.7, Max = 68.1 3) For the data set: {82, 82, 91, 91, 70, 88, 53, 88, 82, 70, 52, 93, 52, 93, 67, 91, 64, 90, 93, 70, 91, 75} 3a. Find the mean. 3b. Find the range. 3c. Find the standard deviation. MEAN = 83.4 RANGE = 99 – 65 = 34 SD = 7.599 4) For the data set: {30, 36, 40, 49, 53, 67, 71, 73, 75, 93} 4a. Find the five number summary. 4c. What is the Interquartile range? IQR = 73 – 40 = 33 MIN = 30, Q1 = 40, M = 60, Q3= 73, MAX= 93 4b. What is the range? 4d. Are any values outliers? NONE; Range = 93 – 30 = 63 40 - 1.5(33) = -9.5 73 + (1.5)33 = 122.5 Honors Discrete Practice Multiple Choice CHAPTER 14: DESCRIPTIVE STATISTICS Pick the MOST ACCURATE Answer Choice For #1 - 8: Refer to the frequency table that shows Joe’s golf scores. Score 2 3 4 5 6 7 8 9 Frequency 5 8 11 5 3 1 2 1 1) The total number of holes Joe played was A. 8 C. 80 B. 36 D. 153 2) Joe’s average golf score was A. 3.75 B. 4 C. 4.25 D. 4.5 3) Joe’s median golf score was A. 3.75 B. 4 C. 4.25 D. 4.5 4) The first quartile of gold scores is A. 2.5 B. 3 C. 3. 5 D. 4 5) The third quartile of golf scores is B. 5.5 A. 5 C. 6 D. 6.5 6) The 95th percentile of golf scores is A. 8 B. 8.2 C. 8.5 D. 34.2 7) The range of golf scores was B. 8 A. 7 C. 9 D. 10 8) What kind of variable is a golf score? A. Qualitative and discrete B. Qualitative and Continuous C. Quantitative and discrete D. Quantitative and Continuous For #9 – 12: Refer to the data set {–1, –3, –5, –7, 0, 0 , 7, 9} 9) #27:The mode of the eight numbers is A. –3.5 B. –0.5 C. 0 D. 4 10)#28:The 15th percentile of the eight numbers is A. –5 B. –4 C. –3 D. –2 11)#29:The standard deviation of the eight numbers is A. 5.172 B. 5.972 C. 7.5 D. 26.75 12) The table below shows the midterm and final exam grades of ten students. Midterm 68 78 92 90 88 82 94 83 71 62 Final Exam 62 77 99 87 85 84 95 98 72 64 Which comparison between the midterm grades and the final exam grades is true? A. The final exam grades have a higher mean and standard deviation than the midterm grades. B. The final exam grades have a lower mean and standard deviation than the midterm grades. C. The final exam grades have a higher mean and a lower standard deviation than the midterm grades. D. The final exam grades have a lower mean and a higher standard deviation than the midterm grades. 13) A baseball team scored the following number of runs in its games this season: 6, 2, 5, 9, 11, 4, 5, 8, 6, 7, 5. There is one more game in the season. If the team wants to end the season with an average of at least 6 runs per game, what is the least number of runs the team must score in the final game of the season? A. 2 B. 4 C. 6 D. 8 Honors Discrete Practice Multiple Choice 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) B C B B A A A C C A A A B CHAPTER 14: DESCRIPTIVE STATISTICS-SOLUTIONS