Statistics

Statistics

Simple univariate statistics

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
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.)
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: $M = 2$
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: $M = 49$
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
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
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
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
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
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
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: $Q_{1} = 10.5, Q_{2} = M = 12, Q_{3} = 14$
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: $Q_{1} = 95, Q_{2} = M = 102, Q_{3} = 112$
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)   $I Q R = 18$ ;     (c)  outlier: 120
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: $Q_{1} = 500, Q_{2} = M = 1000, Q_{3} = 3000$
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

$30 ⩽ w < 40$ 3

$40 ⩽ w < 50$ 7

$50 ⩽ w < 60$ 12

$60 ⩽ w < 70$ 18

$70 ⩽ w < 80$ 10

(a)   Represent the data using a histogram.

(b)   Find the mean.
Solutions:    (b)  Mean: $μ = 60$
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: $μ = 12.8$
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
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
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%
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: $I Q R = 13$
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: $\overset{―}{x} = 22$ (for A and for B);     (b)  Group A: variance: $V a r (x) = 0.96$ , standard deviation: $σ \approx 0.980$ ;   Group B: variance: $V a r (x) = 2.72$ , standard deviation: $σ \approx 1.65$
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: $\overset{―}{x} = 1.8$ ;     (b)  variance: $V a r (x) \approx 1.69$ , standard deviation: $σ \approx 1.30$
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: $\overset{―}{x} = 29.9$ ;     (b)  standard deviation: $σ \approx 0.624$

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

weight (in kg)	frequency
$30 ⩽ w < 40$	3
$40 ⩽ w < 50$	7
$50 ⩽ w < 60$	12
$60 ⩽ w < 70$	18
$70 ⩽ w < 80$	10

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

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

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

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

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

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

Solve the following exercises using the GDC: use OneVar command on TI-nspire and 1-Var Stats command on TI-84.
Click here for help: Univariate statistics using TI-nspire, Univariate statistics using TI-84.

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: $\overset{―}{x} = 5$ , standard deviation: $σ \approx 1.57$ , variance: $σ^{2} \approx 2.45$ ;     (b)  median: $M e d = 5$ , quartiles: $Q_{1} = 4, Q_{2} = M e d = 5, Q_{3} = 6$
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: $\overset{―}{x} = 24$ ;     (b)  standard deviation: $σ \approx 1.98$ , variance: $σ^{2} \approx 3.93$
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: $\overset{―}{x} \approx 50.7$ ;     (b)  standard deviation: $σ \approx 9.36$
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)   $\overset{―}{x} \approx 12.0 in, σ \approx 1.25 in$ ;     (b)   $\overset{―}{x} \approx 30.6 cm, σ \approx 3.18 cm$
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 $x^{\circ} C$ to Farenheit scale using the formula: $y (^{\circ} F) = 1.8 \cdot x (^{\circ} C) + 32$
Solutions:    (a)   $\overset{―}{x} = 25^{\circ} C, σ \approx 3.21^{\circ} C$ ;     (c)   $\overset{―}{x} = 77^{\circ} F, σ \approx 5.77^{\circ} F$

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

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

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

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

Bivariate statistics

Solve the following exercises using the GDC: use TwoVar command on TI-nspire and 2-Var Stats command on TI-84.
Click here for help: Bivariate statistics using TI-nspire, Bivariate statistics using TI-84.

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 $M (\overset{―}{x}, \overset{―}{y})$ .

(d)   Write the equation of the regression line of $y$ on $x$ .
Solutions:    (a)   $\overset{―}{x} \approx 76.3, σ x \approx 9.46; \overset{―}{y} \approx 178, σ y \approx 4.69$ ;     (b)   $r \approx 0.912$ ;     (d)   $y = 0.452 x + 143$
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)   $r \approx 0.648$ ;     (c)   $y = 0.507 x + 2.03$
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)   $y = - 0.00105 x + 4.93$ (Hint: You can use a thousand as a unit. In this case the equation of the trend line is: $y = - 1.05 x + 4.93$ .)
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)   $r \approx 0.962$ ;     (b)   $y = 0.691 x + 38.2$ ;     (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 $[x_{m i n}, x_{m a x}]$

The Spearman's rank correlation coefficient (

r_{S}

ρ

) is equal to Pearson's correlation coefficient, but calculated for ranks and not for values. In order to find it, you need a table of ranks: you must determine who's first (rank 1), who's second (rank 2), who's third (rank 3) etc.

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)   $r_{S} \approx 0.371$

name height weight

Andrew 3 5

Bogdan 1 2

Camillo 2 3

Dietrich 6 6

Émile 4 4

Farouk 5 1
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)   $r_{S} \approx 0.882$

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

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

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

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

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

Index

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

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

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

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

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