Statistics

Simple univariate statistics

1. In the city park there are 40 trees: 12 of them are maple trees, 7 mulberry trees, 15 ginkgos and 6 oaks.
   

   (a)   Represent the data using diagrams of different types.

   (b)   Find the mode.

   Solutions:    (b)  Mode: ginkgo
2. Academic grading system in USA knows grades A, B, C, D and F. Students in a certain class took a math test. Here are their results: 7 students got an A, 5 students got a B, 7 students got a C, 4 students got a D and 2 students got an F.
   

   (a)   Represent the data using diagrams of different types.

   (b)   Find the mode.

   Solutions:    (b)  Modes: A and C. (There are two modes. We can say that the distribution of grades is bimodal.)
3. Teacher asked her students, how many siblings (brothers and sisters) do they have. Here are their answers: Annie 1, Ben 3, Cecilia 0, David 2, Evelyn 1, Fred 4, Ginger 0, Harry 6 and Irene 3.
   

   (a)   Write the data as an ordered list.

   (b)   Find the median.

   Solutions:    (a)  0, 0, 1, 1, 2, 3, 3, 4, 6;     (b)  Median: \(M=2\)
4. During a medical check-up students' weights were recorded. Here are the results: Anastasia 40 kg, Boris 52 kg, Vera 48 kg, Grigory 45 kg, Dasha 52 kg, Evgeny 50 kg, Zhanna 45 kg, Zosim 53 kg, Irina 43 kg, Kostya 52 kg.
   

   (a)   Write the data as an ordered list.

   (b)   Find the median.

   Solutions:    (a)  40, 43, 45, 45, 48, 50, 52, 52, 52, 53;     (b)  Median: \(M=49\)
5. Teacher Lars measured how tall are his students. Here are their heights: Anders 182 cm, Bjørn 195 cm, Christina 180 cm, Dagmar 182 cm, Ebba 176 cm.
   

   (a)   Find the mode.

   (b)   Find the mean.

   Solutions:    (a)  Mode: 182;    (b)  Mean (or average): μ = 183 cm
6. The heights of the players of a basketball team are: 185, 190, 193, 199, 201, 204, 205, 206, 208 cm.
   

   (a)   Find the average (the mean).

   (b)   Find the median.

   Solutions:    (a)  Mean: 199 cm;     (b)  median: 201 cm
7. In a short test students had to answer three questions. The table shows how many students answered 0, 1, 2 or 3 questions.
   

   questions answered	number of students
   0	4
   1	5
   2	6
   3	10

   (a)   Find the mean.

   (b)   Find the mode.

   Solutions:    (a)  Mean: 1.88;    (b)  mode: 3
8. The headmistress conducted a survey. Among many other questions she asked the students about their households. She obtained the following results: 3 students live in a household of 2 members, 6 students live in a household of 3 members, 2 students live in a household of 4 members, 2 students live in a household of 5 members, 2 students live in a household of 6 members.
   

   (a)   Write the data in a table.

   (b)   Find the mean.

   (c)   Find the median.

   Solutions:    (b)  Mean: 3.6 members;     (c)  median: 3 members
9. A statistician checked all the apartments in Penny Lane. He discovered that there are 6 apartments occupied by only 1 inhabitant, 12 apartments occupied by 2 inhabitants, 8 apartments occupied by 3 inhabitants, 7 apartments occupied by 4 inhabitants, 6 apartments occupied by 5 inhabitants and 1 apartment occupied by 6 inhabitants.
   

   (a)   Represent the data using a table.

   (b)   Represent the data using different diagrams.

   (c)   Find the mode, the median and the mean.

   Solutions:    (c)  Mode: 2,   median: 3,   mean: 2.95
10. A teacher asked his students how much time did they spend learning on the previous day. Here are their answers  —  Andrew: 50 minutes, Benjamin: 35 minutes, Claude: 1 hour and 30 minutes, Dolores: 1 hour and 20 minutes, Esmeralda: 2 hours, Frank: 55 minutes, Gloria: 1 hour and 50 minutes, Hans: 20 minutes. Find the average and the median.
    Solutions:    Average or mean: 70 minutes,   median: 67.5 minutes
11. Several contestants participated in a pancake eating contest. Alice ate 12 pancakes, Boris ate 14 pancakes, Connie ate 13 pancakes, Deborah ate 12 pancakes, Edgar ate 14 pancakes, Fernando ate 8 pancakes, George ate 15 pancakes, Helen ate 9 pancakes, Irene ate 15 pancakes, Jeremy ate 11 pancakes, Kim ate 12 pancakes and Louis ate 10 pancakes.
    

    (a)   Find the median and quartiles.

    (b)   Represent the results using box and whisker plot.

    Solutions:    (a)  Median: \(M=12\),   quartiles: \(Q_1=10.5,~ Q_2=M=12,~ Q_3=14\)
12. Draw the box-and-whisker plot for the following data set representing the masses of tomatoes (in grams):
    82, 88, 90, 95, 95, 99, 100, 104, 108, 111, 112, 115, 119, 120.
    Solutions:    Median: \(M=102\),   quartiles: \(Q_1=95,~ Q_2=M=102,~ Q_3=112\)
13. Weights of several persons were measured in kilograms. Here's the ordered list:
    48, 50, 51, 51, 55, 58, 60, 64, 67, 71, 80, 120
    

    (a)   Find the median and the quartiles.

    (b)   Find the interquartile range.

    (c)   Are there any outliers?

    (d)   Represent the results using box and whisker plot.

    Solutions:    (a)  Median: \(M=59\),   quartiles: \(Q_1=51,~ Q_2=59,~ Q_3=69\);     (b)  \(IQR=18\);     (c)  outlier: 120
14. In the Ultrasoft Company the chief executive has a monthly salary of 35 000 €. There are two assistant managers who have a salary of 10 000 €. Four engineers have a salary of 3 000 €. Six foremen have a salary of 1 000 €. The remaining twelve employees earn 500 € monthly.
    

    (a)   Find the mode, the mean and the median.

    (b)   Represent the results using box and whisker plot.

    Solutions:    (a)  Mode: 500 €,   mean: 3 160 €,   median: 1 000 €;     (b)  quartiles: \(Q_1=500,~ Q_2=M=1000,~ Q_3=3000\)
15. The ministry of education ordered that body weights must be measured for all students attending the Charles Darwin Middle School. Results were grouped in classes and the following table of grouped data was obtained:
    

    weight (in kg)	frequency
    \(30\leqslant w\lt 40\)	3
    \(40\leqslant w\lt 50\)	7
    \(50\leqslant w\lt 60\)	12
    \(60\leqslant w\lt 70\)	18
    \(70\leqslant w\lt 80\)	10

    (a)   Represent the data using a histogram.

    (b)   Find the mean.

    Solutions:    (b)  Mean: \(\mu=60\)
16. Students took a math test. The maximum possible number of marks was 20. The following table of grouped data shows their results:
    

    marks	frequency
    0, 1, 2	2
    3, 4, 5	3
    6, 7, 8	1
    9, 10, 11	4
    12, 13, 14	7
    15, 16, 17	5
    18, 19, 20	8

    (a)   Represent the data using a histogram.

    (b)   Find the mean.

    Solutions:    (b)  Mean: \(\mu=12.8\)
17. A farmer measured the lengths of several cucumbers growing in his garden. He obtained the following results (in cm):
    25, 27, 27, 28, 29, 29, 29, 30, 30, 31, 31, 31, 31, 32, 33, 34, 34, 34, 35, 35.
    

    (a)   Write the table of cumulative frequencies.

    (b)   Draw the cumulative frequency diagram.

    Solutions:
    

    cucumber length	cumulative frequency
    25	1
    27	3
    28	4
    29	7
    30	9
    31	13
    32	14
    33	15
    34	18
    35	20

18. The following table shows the results of a math test:
    

    grade	frequency
    1	2
    2	1
    3	3
    4	5
    5	7
    6	8
    7	4

    (a)   Write the table of cumulative frequencies.

    (b)   Draw the cumulative frequency diagram.

    Solutions:
    

    grade	cumulative frequency
    1	2
    2	3
    3	6
    4	11
    5	18
    6	26
    7	30

19. Students measured the shoe sizes in a group of 20 persons. The results are written in the following table:
    

    shoe size	frequency
    37	5%
    38	10%
    39	15%
    40	25%
    41	30%
    42	15%

    (a)   Write the table of cumulative frequencies.

    (b)   Draw the cumulative frequency diagram.

    Solutions:
    

    shoe size	cumulative frequency
    37	5%
    38	15%
    39	30%
    40	55%
    41	85%
    42	100%

20. A statistician measured the weights of 15 students (in kg). The results are written bellow:
    55, 57, 58, 58, 60, 60, 60, 62, 65, 65, 69, 71, 71, 73, 75.
    

    (a)   Draw the cumulative frequency graph.

    (b)   Find the median, the quartiles and the interquartile range.

    Solutions:    (b)  Median: \(M=62\);   quartiles: \(Q_1=58,~ Q_2=M=62,~ Q_3=71\);   interquartile range: \(IQR=13\)
21. The following tables show the distribution of the shoe sizes in two groups of children (labeled Group A and Group B):
    
    Group A:

    shoe size	20	21	22	23	24
    frequency	2	5	10	7	1

    Group B:

    shoe size	20	21	22	23	24
    frequency	8	3	2	5	7

    (a)   Find the mean (for each group separately).

    (b)   Find the variance and standard deviation (for each group separately).

    Solutions:    (a)  Mean: \(\overline{x}=22\) (for A and for B);     (b)  Group A: variance: \(Var(x)=0.96\), standard deviation: \(\sigma\approx0.980\);     Group B: variance: \(Var(x)=2.72\), standard deviation: \(\sigma\approx1.65\)
22. Contestants can win up to 4 points in the Spelling Wasp contest. The following table shows how many contestants won certain number of points:
    

    number of points	0	1	2	3	4
    number of contestants	6	7	8	5	4

    (a)   Find the mean.

    (b)   Find the variance and standard deviation.

    Solutions:    (a)  Mean: \(\overline{x}=1.8\);     (b)  variance: \(Var(x)\approx1.69\), standard deviation: \(\sigma\approx1.30\)
23. In a candy factory candies are produced and packaged in bags. Each bag should contain 30 candies. Internal inspection checked 20 bags: one bag contained only 28 candies, two bags contained 29 candies, two bags contained 31 candies and all the other bags contained 30 candies.
    

    (a)   Find the mean.

    (b)   Find the standard deviation.

    Solutions:    (a)  Mean: \(\overline{x}=29.9\);     (b)  standard deviation: \(\sigma\approx0.624\)

24. The following table shows the results of a math test:
    

    grade	frequency
    1	1
    2	2
    3	2
    4	5
    5	7
    6	9
    7	5

    (a)   Find the mean, the standard deviation and the variance.

    (b)   Find the median and the quartiles.

    Solutions:    (a)  Mean: \(\overline{x}=5\), standard deviation: \(\sigma\approx1.57\), variance: \(\sigma^2\approx2.45\);     (b)  median: \(Med=5\), quartiles: \(Q_1=4, Q_2=Med=5, Q_3=6\)
25. At gym several students were trying to do as many push ups as possible in one minute. The following table shows their results:
    

    push ups	frequency
    21	4
    22	3
    23	8
    25	10
    27	4
    28	1

    (a)   Find the mean.

    (b)   Find the standard deviation and the variance.

    Solutions:    (a)  Mean: \(\overline{x}=24\);     (b)  standard deviation: \(\sigma\approx1.98\), variance: \(\sigma^2\approx3.93\)
26. Policemen controlled the traffic on the main street in their town. They were measuring speeds of cars passing by. The following table shows the results they obtained (in km/h):
    

    speed	frequency
    45	3
    47	8
    48	15
    49	6
    50	32
    51	12
    52	9
    76	1
    132	1

    (a)   Find the mean.

    (b)   Find the standard deviation.

    Solutions:    (a)  Mean: \(\overline{x}\approx50.7\);     (b)  standard deviation: \(\sigma\approx9.36\)
27. An american fisherman measured the lengths of the fish he caught today. His results are given in inches:
    

    inches	frequency
    10	4
    11	6
    12	9
    13	8
    14	2
    15	1

    (a)   Find the mean and standard deviation of the data given above.

    (b)   Convert inches to centimetres and find the mean and standard deviation in centimetres.

    Hint:    1 inch = 2.54 cm
    Solutions:    (a)   \(\overline{x}\approx12.0~\mathrm{in},~\sigma \approx1.25~\mathrm{in}\);     (b)   \(\overline{x}\approx30.6~\mathrm{cm},~\sigma \approx3.18~\mathrm{cm}\)
28. Dagmar lives in Europe and she is a meteorologist. She recorded the temperature at noon for each day in a week. Here are her results:
    

    Day	Sun	Mon	Tue	Wed	Thu	Fri	Sat
    Temperature (°C)	21	24	26	30	29	23	22

    (a)   Find the mean and standard deviation of her data.

    Dagmar e-mailed her data to David who is an american meteorologist. Americans use degrees Farenheit (°F) for measuring the temperature.

    (b)   Convert Dagmar's data to degrees Farenheit.

    (c)   Find the mean and standard deviation in degrees Farenheit.

    Hint:    Convert \(x~^\circ\mathrm{C}\) to Farenheit scale using the formula: \(y(^\circ\mathrm{F}) = 1.8\cdot x(^\circ\mathrm{C})+32\)
    Solutions:    (a)   \(\overline{x}=25~^\circ\mathrm{C},~ \sigma \approx3.21~^\circ\mathrm{C}\);     (c)   \(\overline{x}=77~^\circ\mathrm{F},~ \sigma \approx5.77~^\circ\mathrm{F}\)

Bivariate statistics

29. The following table shows the weights and heights of 6 students.
    

    weight (kg)	61	70	72	82	85	88
    height (cm)	170	178	175	181	179	185

    (a)   Find the mean and standard deviation for weight \((x)\) and for height \((y)\) .

    (b)   Find the Pearson product-moment correlation coefficient \(r\).

    (c)   Draw scatter graph. Include the mean point \(M(\overline{x},\overline{y})\).

    (d)   Write the equation of the regression line of \(y\) on \(x\).

    Solutions:    (a)  \(\overline{x}\approx76.3,~\sigma x\approx9.46;~~ \overline{y}\approx178,~ \sigma y\approx4.69\);     (b)  \(r\approx0.912\);     (d)  \(y=0.452x+143\)
30. The following table shows the grades of 8 students in mathematics and in physics.
    

    mathematics	7	4	6	5	3	6	1	5
    physics	5	5	4	4	2	7	3	5

    (a)   Draw scatter graph.

    (b)   Find correlation coefficient \(r\).

    (c)   Write the equation of the line of best fit (regression line).

    Solutions:    (b)  \(r\approx0.648\);     (c)  \(y=0.507x+2.03\)
31. The following table shows monthly incomes and number of children for several families.
    

    incomes	1200	1800	2100	2500	2800	3500	3800
    children	4	3	2	2	3	1	1

    (a)   Draw scatter graph.

    (b)   Find the correlation coefficient \(r\).

    (c)   Draw the trend line (regression line) and write its equation.

    Solutions:    (b)  \(r\approx-0.869\);     (c)  \(y=-0.00105x+4.93\) (Hint: You can use a thousand as a unit. In this case the equation of the trend line is: \(y=-1.05x+4.93\).)
32. A statistician is conducting a survey on mock exams and final exams. Here's his data for six students: results of mocks and finals are written in percent (marks 0 to 100).

    student	A	B	C	D	E	F
    \(x=\) mock exams	31	44	56	59	73	80
    \(y=\) final exams	59	69	72	85	89	92

    (a)   Find the correlation coefficient \(r\).

    (b)   Write the equation of the regression line of \(y\) on \(x\).

    Make the following predictions if possible:

    (c)   A student got 49 marks on mocks. Predict his marks on finals.

    (d)   A student got 62 marks on mocks. Predict her marks on finals.

    (e)   A student got 95 marks on mocks. Predict her marks on finals.

    Solutions:    (a)  \(r\approx0.962\);     (b)  \(y=0.691x+38.2\);     (c)  He'll get 72 marks on finals.;     (d)  She'll get 81 marks on finals.;     (e)  Prediction not possible. The value 95 is not on the interval \(\left[x_{min},x_{max}\right]\)

33. Heights (cm) and weights (kg) of six persons were measured.
    

    name	height	weight
    Andrew	178	75
    Bogdan	190	84
    Camillo	185	82
    Dietrich	170	72
    Émile	177	80
    Farouk	172	85

    (a)   Write the table of ranks for height and weight.

    (b)   Find the Spearman's rank correlation coefficient \(r_S\).

    Solutions:    (b)  \(r_S\approx0.371\)

    name	height	weight
    Andrew	3	5
    Bogdan	1	2
    Camillo	2	3
    Dietrich	6	6
    Émile	4	4
    Farouk	5	1

34. Heights (cm) and weights (kg) of six persons were measured.
    

    name	height	weight
    Annie	169	64
    Bojana	182	69
    Chiara	178	64
    Dagmar	163	48
    Éloïse	172	60
    Fatima	163	56

    (a)   Write the table of ranks for height and weight.

    (b)   Find the Spearman's rank correlation coefficient \(r_S\).

    Solutions:    (b)  \(r_S\approx0.882\)

    name	height	weight
    Annie	4	2.5
    Bojana	1	1
    Chiara	2	2.5
    Dagmar	5.5	6
    Éloïse	3	4
    Fatima	5.5	5

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