# CHAPTER 2 DESCRIPTIVE MEASURES

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1 CHAPTER 2 DESCRIPTIVE MEASURES Decide whether the following statements are true (T) or false (F). The arithmetic mean always describes the distribution of data well. The median and the mode can always be unambiguously determined. The standard deviation is actually the square mean of the data. Standard deviation and dispersion are similar concepts. The coefficient of variance measures the differences of the data in percentage form. The A and F measures of skewness always have the same negative or positive sign. The values of the fifth fractile and the second quartile are always the same. The K measure of concentration ranges between [-1;+1]. The more spread out the data, the smaller the concentration. Measures of central tendency can only be calculated in case of quantitative variables. Test questions (There is only one right answer to each question): 1. If the average depth of a lake is 1.4 meters, it means that A. there could be a spot in the lake where it is deeper than 1,4 meters. B. an adult of average height can walk through the lake. C. the deepest point of the lake is 1.4 meters. 2. Which measure would you use to describe the most frequent summer temperature? A. Median B. Mode C. geometric mean 3. Which statement is NOT true? A. Not only quantitative variables have a mode. B. The arithmetic mean can be calculated from any variables. C. The median can always be determined from quantitative variables. 4. A Schwäbischen pretzel costs 0.5 euro in most of the bakeries in Stuttgart. Then A. 0.5 euro is the mean. B. 0.5 euro is the median. C. 0.5 euro is the mode. 5. In the canteen the guests paid 520 HUF in the most cases, but half of the guests paid even more. Then. A. Mo = 520 HUF; Me 520 HUF B. Mo = Me = 520 HUF C. Mo = Me 520 HUF - 1 -

2 6. F means: A. the ratio of standard deviation and the arithmetic mean. B. strength of the relation between qualitative and quantitative variables. C. the skewness of the middle 50% of the elements % of middle-sized companies paid their tax obligations by deadline. Then this 25% as A. a typical value. B. a quartile. C. a distribution ratio. 8. A kilo of French salad costs 4 euros in most places in France. Then A. 4 euros is the mean. B. 4 euros is the mode. C. 4 euros is the maximum price. 9. At the beginning of the year every employee gets a 10% pay raise at a company with 10 employees. Then A. the average salary and the standard deviation do not change. B. the average salary grows by 10 10% and the standard deviation does not change. C. both the average salary and the standard deviation grow by 10 %. 10. If 75% of the students in a student group get more than 45 scores, then A. the average score is more than 45. B. the upper quartile is 45. C. the lower quartile is σ means: A. the square sum of individual differences B. the squared average of the deviations from the mean C. the geometric mean of the variance % of the unemployed in Nógrád county searched for new jobs for at least 5 months. Then A. Q 1 = 25 B. Me = 25 C. Q 3 = A kilo of salami costs at least 5 euros in Paris. Then A. 5 euros is the mean. B. 5 euros is the minimal value. C. 5 euros is the standard deviation. 14. The suburban railway runs every 8 th minute on average with a 2 minute standard deviation. Then. A. the coefficient of variance is 25%. B. the range is 4 minutes. C. CV = 4 %

3 15. If the test results disperse in the interval of 25 and 49, then. A. the standard deviation is at most = 24. B. the median equals (49 25)/2 = 12. C. the mean is smaller than Passengers must wait for the suburban railway more than 5 minutes in a quarter of the cases. Then this 5 minutes is equal to. A. the lower quartile. B. the mode. C. the upper quartile. 17. Which could be the standard deviation of the following five test results: 11 scores, 13 scores, 14 scores, 18 scores, and 19 scores? A. 91% B. 19 scores C. 1,9 scores 18. According to the Hungarian Statistical Office the average monthly price change was 0.8% between January and May. Then from January to May the prices in total A. grew by %, that is by 4 %. B. increased by =1.47, that is by 47 %. C. grew by =1.0406, i.e. by 4.06 %. 19. The F measure A. is always smaller than the A measure. B. is one of the measures of skewness. C. value ranges between 0 and In case of right-skewed distribution: A. Mo < x B. Mo > x C. Mo = x 21. The suburban railway arrives within five minutes in half of the cases. Then. A. Me 5 B. Me 5 C. Me = The average of five students test results equals 50, with a standard deviation of 5 scores. Then A. CV = 5 % B. CV = 10 % C. CV = 0.1 % 23. Which statement is TRUE? A. In case of quantitative variables, only the arithmetic mean can be calculated. B. The median can always be determined in case of quantitative variables. C. The mode can always be determined in case of discrete/individual data

4 24. If prices grow per month by 0.2% on average, then at yearly level, it means A. a rate of price increase. B rate price increase. C rate price increase. 25. Most of the students got 18 scores on the Statistics test but half of the students made at least 17 scores. Then A. Mo = 18 scores Me 17 B. Mo 18 scores, Me 17 C. Mo = 18 scores, Me = If, in a group of employees with average earnings of HUF per month, everybody gets a HUF pay raise, then.. A. the average salary and the standard deviation will not change. B. the average salary and the median will not change. C. the average salary and the median grow by 10 thousand HUF. 27. Not more than 0.5 euros must be paid for a Schwäbischen pretzel in Stuttgart. Then A. 0.5 euros is the maximum price. B. 0.5 euros is the median. C. 0.5 euros is the standard deviation. 28. If the price of a commodity grows by 6% from January to June, then per month on average A. the price grew by 6 6 percentage. B. the price changed up to 1.06/6 times the original price. C. the price changed up to times the original price. 29. In case of right-skewed distribution, A. Q 3 Me > Me Q 1 B. Q 3 Me = Me Q 1 C. Q 3 Me < Me Q The criterion of the central position of the measures of central tendency is. A. Mo x Me B. the measures of central tendency are always between the highest and the lowest values. C. a measure of central tendency can never be greater than the range

5 Exercises Exercise 1 In order to increase the number of customers, a fast-food restaurant regularly monitors its times of services to ensure that its speed of service is improving in time. A sample of 20 times of service has been taken on a randomly selected day and is shown here (time in seconds): 45, 48, 49, 56, 61, 66, 66, 66, 70, 72, 72, 75, 78, 79, 81, 81, 83, 95, 102, 135; a) Determine the average, the most frequent and the median time of service. b) Compute and interpret the known measures of dispersion. c) Calculate and interpret the quartiles. d) Characterise the skewness with the calculation of the relevant coefficients. Exercise 2 In order to check the operation of a 250 gram coffee filling and packaging machine, a sample of 100 coffee packages were examined on a randomly selected day: Filling weight of the packages (grams) Number of packages (pieces) less than but less than but less than but less than or more 10 Total 100 Question: a) Compute and interpret the common measures of centre and location. b) Calculate the measures of Dispersion (IQR (Interquartile Range), MAD (Mean Absolute Deviation), standard deviation, variance, Coefficient of Variation) and interpret their meaning based on the case. c) Calculate and interpret the different coefficients of skewness. Exercise 3 In a higher educational institution the comprehensive applied mathematics exam is comprised of two parts. On the first day, 20 students took the exam, the results of which are presented below: Oral exam results: 4, 1, 4, 5, 3, 2, 3, 4, 3, 5, 2, 2, 4, 3, 5, 5, 1, 1, 1, 2. Written exam results: 2, 3, 1, 4, 2, 5, 3, 1, 2, 1, 2, 2, 1, 1, 2, 3, 1, 2, 3, 4. a) Compile a statistical table based on the data above. b) Calculate the mean, the mode, the median, the standard deviation and the coefficient of variation of the oral and written exams separately and together as well. c) Compare the quartile distribution and then indicate whether the data are skewed or symmetric. d) Report on your findings in writing

6 Exercise 4 The following data are known about students buying habits in the college canteen on a given day. every tenth student spent less than 200 HUF; students paying at least 800 HUF spent HUF altogether in the canteen; 20% of the students paid at least 600 HUF. Amount of money spent Number of customers (HUF) (f i ) cf i g i (%) cg i (%) s i under but under but under but under and above Total 600 a) Fill in the above table with the help of the information given above. b) Evaluate the distribution of the purchasing value with the known measures of central tendency and dispersion and with the coefficients of skewness. Exercise 5 50% of the employees of a company are over 34, and 25% of them are over 44 years old. Half of the employees, who are younger than 34, is also under the age of 28. Question: Calculate the F measures of skewness of age and interpret the result. Exercise 6 A firm employs 75 people on one of its premises, who do the same assembling job. On a given day the workers performance was spread between 70 and 95, and on average they assembled 80 pieces. Half of the workers assembled more than 84, while most of the workers assembled 89. During a quality control it turned out that the best performing worker assembled 6 defective pieces, which should not have been counted in the total. a) After subtracting the defective pieces, what will the following indicators be? Average performance=. Mode= Median= b) After subtracting the defective pieces, what will the indicators be if the quality control shows that the worst performing worker assembled 6 defectives pieces? Exercise 7 The following data and information are known about the sum of the regular social allowances and the number of people getting these allowances in a village in the year The average sum of social allowances per capita equalled HUF, from which the sum of allowances on average deviated by 45 %. The most people got approximately HUF. Half of the people living on allowance received less than HUF. The middle 50% of the allowances was skewed strongly to the left

7 On the basis of the data in the table below, examine how the distribution of the regular social allowances per capita in the village changed in the year Support your answers by calculating the relevant indicators. Sum of social allowance per capita in the year 2004 (thousand HUF/capita) Number of people on allowance (person) under but less than but less than but less than or more 20 Total 300 Exercise 8 The distribution of the motor vehicles of a domestic transportation company based on the kilometres run has been the following: There are only 2 such vehicles that have run less than kms; Every fifth vehicle has run at least kms; Vehicles running at least kms but less than kms have run kms in total; There are 16 such vehicles in the company that have already run at least kms. Kilometres run (km) Number of vehicles (f i ) cf i g i s i less than but less than but less than but less than but less than or more Total 40 a) Fill in the table based on the information given above. b) With the help of the learned measures of distribution, evaluate the measures of central tendency and of dispersion, the measures of skewness, and the distribution of the kilometres run by the vehicles. Exercise 9 The average salary at a company equalled HUF with a HUF standard deviation, half of the workers earned at most HUF while most workers earned HUF. a) If there is a fix HUF pay rise during the following month, then the average salary will be. HUF, with a. HUF standard deviation while the mode will be HUF and the median will be HUF

8 b) If in the following month there is a 10 % pay rise then the average salary will be HUF, with a HUF standard deviation while the mode will be HUF and the median will be HUF. c) If in the following month there is a fix HUF pay rise and then a further 5% pay rise then the average salary will be. HUF, with a. HUF standard deviation while the mode will be HUF and the median will be HUF. Exercise 10 In a college the results of a statistics test are the following: 22% of the students received scores in the range of 30 35; only every eighth student received less than 10 scores; Final scores Number of (scores) students (f i ) cf i g i (%) cg i (%) under but under but under but under or above 65 Total 400 a) Fill in the above table based on the information given above. b) Evaluate the distribution of the test results with the help of the known measures of distribution measures of central tendency and of variation and the measure of skewness. Exercise 11 The following sentences are in the 2004 report of a travelling agency: Our guests are of the age of 34 on average, but half of the guests were younger than 30. Most of our guests were children of age around 13 with parents and grandparents. Our oldest guest was 72 years old, the age of our guests spread out significantly by 40% around the average age. a) Comment on the above statements based on the table below and correct the analysis if necessary. Support your comments with the calculation of adequate measures. Age groups of guests (year) Number of guests (person) younger than but younger than but younger than but younger than but younger than or older 10 Total 240 b) Which further measures of centre and location would you recommend to calculate? Why? - 8 -

9 Exercise 12 The following table presents how the number of enterprises in the agricultural and food industry sectors changed in 2004, grouped by the number of employees: Number of employees (person) Number of agricultural enterprises (pieces) Number of food industrial enterprises (pieces) 0 but fewer than but fewer than but fewer than but fewer than but fewer than or more Total Source: Hungarian Central Statistical Office Compare, with the help of the measures of central tendency, dispersion and skewness, how the number of agricultural and food industrial enterprises changed with respect to the number of their employees. Exercise 13 A passenger transportation enterprise operates two microbuses which consumed 45 and 72 liters of petrol a day, while their average consumption is 15 and 18 liters/100 kms, respectively. Question: What is the combined average consumption of the two microbuses? Exercise 14 The distribution of the waiting time patients must spend in a health centre during a given week was the following: One quarter of the patients waited minutes in the waiting room of the health centre; 30% of the patients had to wait at least 35 minutes; Waiting time Number of (minutes) patients (f i ) cf i g i (%) cg i (%) less than but less than but less than but less than but less than or more Total a) Fill in the above table based on the information given above. b) Evaluate the distribution of the waiting time with the help of the known measures of distribution, measures of centre and location and of dispersion and the measure of skewness

10 Exercise 15 The results of the final test in Statistics I. can be seen below: 35% of the students got less than 80% however 40% of them made at least 90% 20% of the students got less than 70%. Test result (%) Number of students (f i ) cf i g i (%) cg i (%) < but < but < but < but < 100 Total 440 a) Fill in the table based on the information given above. a) Evaluate the distribution of the test results with the help of the known measures of distribution, measures of central tendency and of dispersion and the measure of skewness. Exercise 16 The following table shows the figures of the monthly average salary at a company in The management plans to give an average pay raise of HUF to both blue and white collar workers in 2005, but, depending on the number and rate of the workers, the pay raise could result in a different monthly average salary: Number (person) a.) 2005 b.) 2005 c.) Average Average Average Number Number Number salary salary salary (person) (person) (person) (HUF/head) (HUF/head) (HUF/head) Average salary (HUF/head) Blue collar White collar Total Determine the employee mix in a way that, based on the higher salary in 2005, the average salary of the whole company would be a.) smaller than in 2004; b.) higher than in 2004; c.) the same as in Exercise 17 The grouped frequency distribution of the firms in the processing industry according to the numbers of employees in Hungary in 2004 is given in the following table: Number of employees (people) Distribution of operating firms (%) Total Source: Hungarian Central Statistical Office

11 a) Determine the average and the most frequent number of employees in the processing industry. b) Calculate and interpret the standard deviation and the coefficient of variation. c) Report on the quartile distribution in writing. Exercise 18 The following information is known about the motor vehicles stopping at a petrol station on a given day to fill up their petrol tanks: 20% of the drivers filled less than 10 litres; The same number of drivers filled at least 15 but less than 20 litres of petrol; Only 75 drivers filled at least 25 litres; 65% of the drivers bought less than 20 litres of petrol. Number of Volume of petrol bought (l) customers (f i ) less than but less than but less than but less than but less than or more Total 400 cf i ' g i (%) cg i '(%) a) Fill in the table using the information given above. b) Report on the distribution of the volume of the purchased petrol with the help of the learned measures of distribution measures of centre, location, of dispersion and the measures of skewness. Exercise 19 A travelling agency organised two types of packaged tours last year. The first type gave the company 36% of its revenue, and they cost HUF on average. The second type on average cost HUF. Question: How much did the packaged tours cost on average at this company? Exercise 20 In an assembling plant 70 people work, 40 of them male and 30 female. When examining the productivity of the plant, the following data were revealed about the performance of the males in a given month: Number of assembled pieces Number of workers (person) Total

12 It is also known that females assembled pieces altogether, and the square sum of their performance equalled a) Using the data in the table above, characterise the distribution of the productivity of males with the help of measures of central tendency. b) Determine and interpret the MAD (Mean Absolute Deviation), the standard deviation, and the variance as well as the coefficient of variation. c) Determine the quartile distribution of the productivity of males. d) Describe the shape of the distribution (skewness). e) Compare the productivity of the males and the females with the help of the available measures of central tendency, of dispersion, and shape. Exercise 21 A company s Human Resource department did a survey about the times employees spend on coming to work in the morning. The head of the department, after having examined the data gathered from the employees, thinks that there must be a mistake in the survey. Question: a) Examine whether the data below underpin the highlighted statements of the report. The average travelling time equalled 25 minutes, and the travelling times of the individual employees were spread out by 45% on average. Most of the employees travelled 35 minutes. Half of the employees travel at least 38 minutes. Average travelling time (minutes) Number of employees (person) Total 300 b) What further measures would you recommend in order to give a more detailed description of the travelling times? Question 22 The following table displays the number and the income estimates of the non-profit organizations in 2002: Income (thousand HUF) Number of organizations (pieces) Income estimate (million HUF) < but < but < but < but < but < or above Total Source: Hungarian Central Statistical Office

13 Question: a) Characterise the concentration and draw the Lorenz curve. b) Determine the rate of concentration by calculating the coefficient of concentration. Exercise 23 The following table displays the price revenues of a domestic hotel chain in 2004: Revenues (million HUF) Number of hotels (pieces) Total 24 Question: Based on the data above decide whether the following statements are true or false (T/F). Support your decision with the calculation of the relevant measures.. ~ the average income of the hotels was 225 million HUF with a standard deviation of 18%;. ~ the distribution of the middle 50% of the hotels is strongly skewed to the right;. ~ the typical income equalled 175 million HUF. Exercise 24 Lately the walls of several houses cracked in a village. According to the inhabitants the phenomenon appeared due to the recently increased traffic. They intended to support their statement with data since they counted the motor vehicles during the last month (30 days). Unfortunately, the table presenting the data as well as the results had partly been burnt before the written report was prepared and submitted to the local government. Only the following pieces of information could be saved: Intervals of equal length were used in the grouping; there was only one single day when more than motor vehicles crossed the examined road segment; every fifth day more than 500 but not more than motor vehicles were counted; mo = 1 000; k 1 = k 2 = 6; on average motor vehicles were counted daily on the examined road segment. Number of motor vehicles crossing Number of days Total

14 a) Reconstruct the original table with the help of the data saved. b) After calculating the adequate measures of central tendency, dispersion and shape, report in writing how the crossing traffic changed. Exercise 25 At a company 5% of the blue collar workers earn under 350 HUF/hour, 48 workers earn between 350 and 399 HUF/hour, 35% of them get between 400 and 499 HUF/hour, 37 employees receive between 500 and 599 HUF/hour, and finally 12% of the blue collar workers earn between 600 and 799 HUF/hour. There are 200 blue collar workers employed by the firm. a) Based on the given information tabulate a grouped frequency distribution. b) Calculate and interpret the coefficient of variation and the F coefficient of skewness. Exercise 26 The weights of the suitcases and other packages put on a MALÉV flight were the following: Weight (kg) Number of luggage under but under but under but under but under but under or above 13 Total 170 Report on the luggage in writing according to the distribution of their weight with the help of the known measures of central tendency and dispersion and the measure of skewness. Exercise visitors turned up in the city during the first month of the academic year, who borrowed books altogether. Question: Determine and interpret the coefficient of variation if the square sum of the borrowed books is Exercise 28 The following graphs present how the number and the gross size of the premises of the shopping malls and the hypermarkets changed between 1996 and 2003:

15 Open premises at the end of the year Shopping malls Hypermarkets Gross size of the premises (sqm) at the end of the year Shopping malls Hypermarkets Source: Association of Hungarian Shopping Malls Question: With the calculation of adequate measures compare how shopping malls and hypermarkets spread throughout Hungary in the given time period

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