14 8 9 4 0 Example 1. The number of goals scored by a team in 20 games are given below : 3, 2, 4, 2, 2, 3, 2, 2, 0, 5, 1, 1, 2, 3, 0, 2, 1, 4, 1, 0 Mean.

14

8

9

4

0

Example 1.The number of goals scored by a team in 20 games are given below :3 , 2 , 4 , 2 , 2 , 3 , 2 , 2 , 0 , 5 , 1 , 1 , 2 , 3 , 0 , 2 , 1 , 4 , 1 , 0

Mean from a Frequency Table

f.xFrequency, fGoals x

Calculating the Mean: If there are large amounts of data, it is easier if it is displayed in a frequency table.

0

1

2

3

4

5

3

4

7

3

2

1 5

∑f= 20 ∑fx= 40

Mean =∑fx

∑f

= 40

20= 2

Mode

14

19

17

7

3

Example 1.The number of goals scored by a team in 20 games are given below :3 , 2 , 4 , 2 , 2 , 3 , 2 , 2 , 0 , 5 , 1 , 1 , 2 , 3 , 0 , 2 , 1 , 4 , 1 , 0

Median from a Frequency Table

C. F.Frequency, fGoals x

Calculating the median If there are large amounts of data, it is easier if it is displayed in a frequency table.

0

1

2

3

4

5

3

4

7

3

2

1 20

∑f= 20Mode

(20)/2= 10

(20)/2 + 1

= 11

The 10th value is 2The 11th value is 2∴ MEDIAN = ( 2+2 ) / 2 = 4/2 = 2

Large quantities of data can be much more easily viewed and managed if placed in groups in a frequency table. Grouped data does not enable exact values for the mean, median and mode to be calculated. Alternate methods of analyising the data have to be employed.

Example 1.During 3 hours at Heathrow airport 55 aircraft arrived late. The number of minutes they were late is shown in the grouped frequency table below.

Grouped Data

250 - 59

440 - 49

530 - 39

720 - 29

1010 - 19

270 - 9

frequencyminutes lateData is grouped into 6 class intervals of width 10.

Example 1.During 3 hours at Heathrow airport 55 aircraft arrived late. The number of minutes they were late is shown in the grouped frequency table below.

Grouped Data

midpoint(c.c.)

F × c.c.

250 - 59

440 - 49

530 - 39

720 - 29

1010 - 19

270 - 9

Frequency,fminutes Late

Estimating the Mean: An estimate for the mean can be obtained by assuming that each of the raw data values takes the midpoint value of the interval in which it has been placed.

4.5

14.524.534.5

44.5

54.5

121.5

145

171.5172.5

178

109

Mean estimate = 897.5/55 ≈ 16.32 minutes

55f 5.897..ccf

Example 1.During 3 hours at Heathrow airport 55 aircraft arrived late. The number of minutes they were late is shown in the grouped frequency table below.

Grouped Data

The Modal Class

250 - 59

440 - 49

530 - 39

720 - 29

1010 - 19

270 - 9

frequencyminutes late

The modal class is simply the class interval of highest frequency.

Modal class = 0 - 9

For the following set of data find : أوجد التالية البيانات لمجموعة : بالنسبة

worksheet

الوسيط (2

المنوال (3

المدى (4

الحسابي (1 المتوسط6 , 8 , 5 , 11 , 3 , 1 , 7 , 9 , 3

1) The mean

2) The median

3) The mode

4) The range

أوجد التالية البيانات لمجموعة : For the following set of data find: بالنسبة

9 , 3 , 8 , 7 , 1 , 9 , 11 , 4 , 3 , 2

الوسيط (2

المنوال (3

المدى (4

الحسابي (1 The mean (1المتوسط

2) The median

3) The mode

4) The range

The ages of a random sample of 30 persons are given in the table :

Age ( x ) ( f ) X . f ( c f )

40 2

41 7

42 9

43 6

44 5

45 1

total

من عشوائية عينة كما 30أعمار شخص: بالجدول

: أوجد

worksheet

الوسيط (2 لألعمار

المنوال (3

مدى (4األعمار

الحسابي (1 المتوسطلألعمار

Find :

1) The mean age

2) The median

3) The mode

4) The range of the ages

The following frequency distribution represents the lengths of 20 persons

intervals ( f ) c.c. c.c.× f

150 - 154 2

155 - 159 8

160 - 164 5

165 - 169 4

170 - 179 1

total

أطوال يمثل التكراري شخص 20التوزيع

: أوجد

worksheet

وقدر (2 المنوالية الفئة اكتب المنوال

مدى (3 أوجداألطوال

الحسابي (1 المتوسطلألطوال

Find :

1) The mean of the lengths

2) The model class and estimate the mode

3) The range of the lengths

The grades of 25 students are given below :

Intervalsالفئات

( f ) ( c f )

total

يلي طالب 25درجات كما :

worksheet

التكراري (2 المضلع ارسمقيمة (3التراكمي لتقدر الرسم استخدمالوسيط

فئات (1 ذو تكراري جدول في الدرجات ضع1) Put the grades in a frequency table with intervals

2) Draw the cumulative frequency polygon

3) Use the graph to estimate the median

42 , 63 , 47 , 77 , 46 , 71 , 68 , 83 , 91 , 55 , 67 , 66 , 63 , 57 , 50 , 69 , 73 , 82, 77 , 58 , 66 , 79 , 88 , 97 , 86

(55+1)/2= 28

Example 1.During 3 hours at Heathrow airport 55 aircraft arrived late. The number of minutes they were late is shown in the grouped frequency table below.

Grouped Data

The Median Class Interval

The Median Class Interval is the class interval containing the median.

250 - 59

440 - 49

530 - 39

720 - 29

1010 - 19

270 - 9

frequencyminutes late

The 28th data value is in the 10 - 19 class

Data is grouped into 8 class intervals of width 4.

Example 2.A group of University students took part in a sponsored race. The number of laps completed is given in the table below. Use the information to: (a) Calculate an estimate for the mean number of laps.(b) Determine the modal class.(c) Determine the class interval containing the median.

Grouped Data

136 - 40

231 – 35

2526 – 30

1721 – 25

2016 – 20

1511 – 15

96 – 10

21 - 5

frequency (x)

number of laps

c.c. X fmidpoint(c.c)

136 - 40

231 – 35

2526 – 30

1721 – 25

2016 – 20

1511 – 15

96 – 10

21 - 5

frequencynumber of laps

Example 2.A group of University students took part in a sponsored race. The number of laps completed is given in the table below. Use the information to: (a) Calculate an estimate for the mean number of laps.(b) Determine the modal class.(c) Determine the class interval containing the median.

Grouped Data

3

8

1318

23

2833

38

6

72195360

39170066

381828fx 91f

Mean estimate = 1828/91 = 20.1 laps

Modal Class 26 - 30

Grouped Data

Example 2.A group of University students took part in a sponsored race. The number of laps completed is given in the table below. Use the information to: (a) Calculate an estimate for the mean number of laps.(b) Determine the modal class. (c) Determine the class interval containing the median.

136 - 40

231 – 35

2526 – 30

1721 – 25

2016 – 20

1511 – 15

96 – 10

21 - 5

frequency (x)

number of laps

136 - 40

231 – 35

2526 – 30

1721 – 25

2016 – 20

1511 – 15

96 – 10

21 - 5

frequency (x)

number of laps

Example 2.A group of University students took part in a sponsored race. The number of laps completed is given in the table below. Use the information to: (a) Calculate an estimate for the mean number of laps.(b) Determine the modal class. (c) Determine the class interval containing the median.

Grouped Data

(91+1)/2 = 46

91f

The 46th data value is in the 16 – 20 class , median ≈ 18

c. F.

2

11

26

46

63

88

90

91