Data Handling
Data: A group of observations is called data. A data gives us many information, if it is organized in proper way.
Average: A number which represents the central tendency of a group of observations is called average.
Arithmetic Mean: This is a synonym of average. For calculating the average or arithmetic mean of a set of data, you need to use following formula:
`text(Mean)=text(Sum of each observation)/text(Number of observations)`
Mode: In a set of observations, a particular observation may occur most number of times. This particular observation is called the mode.
Median: The value which lies in the middle of the data so that half of the observations are above it and the other half are below it.
Mean, median and mode are different forms of central tendency of a data. The central tendency of a data shows the representative value of a data.
Exercise 3.1
Question 1: Find the range of heights of any ten students of your class.
Answer: Height of 10 students of my class (in cm) = 145, 144, 148, 157, 140, 139, 142, 140, 138, 146
Here; lowest number = 138
Highest number = 157
So, range of height = Tallest height – Shortest height
= 157 cm – 138 cm = 19 cm
Question 2: Organise the following marks in a class assessment, in a tabular form.
4, 6, 7, 5, 3, 5, 4, 5, 2, 6, 2, 5, 1, 9, 6, 5, 8, 4, 6, 7
Answer: These numbers can be arranged in ascending order as follows:
| Marks | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 7 | 7 | 8 | 9 |
(a) Which number is the highest?
Answer: Highest number = 9
(b) Which number is the lowest?
Answer: Lowest number = 1
(c) What is the range of the data?
Answer: Range of data = Highest marks – Lowest marks
`= 9 – 1 = 8`
(d) Find the arithmetic mean.
Answer: Arithmetic mean can be calculated as follows:
`text(Mean)=text(Sum of each observation)/text(Number of observations)`
=(1 + 2+ 2 + 3 + 4 + 4 + 4 + 5 + 5 + 5 + 5 + 5 + 6 + 6 + 6 + 7 + 7 + 8 + 9) ÷ 20
`=(1+4+3+12+25+18+14+8+9)/(20)`
`= (94)/(20) = 4.7`
Question 3: Find the mean of the first five whole numbers.
Answer: First five whole numbers = 0, 1, 2, 3, 4
`text(Mean)=text(Sum of each observation)/text(Number of observations)`
`=(0+1+2+3+4)/(5)`
`=(10)/(5)=2`
Question 4: A cricketer scores the following runs in eight innings: (58, 76, 40, 35, 46, 45, 0, 100.) Find the mean score.
Answer: Mean can be calculated as follows:
`text(Mean)=text(Sum of each observation)/text(Number of observations)`
=(58 + 76 + 40 + 35 + 46 + 45 + 0 + 100) ÷ 8
`=(400)/(8)=50`
Question 5: Following table shows the points of each player scored in four games:
| Player | Game 1 | Game 2 | Game 3 | Game 4 |
|---|---|---|---|---|
| A | 14 | 16 | 10 | 10 |
| B | 0 | 8 | 6 | 4 |
| C | 8 | 11 | Did not play | 13 |
Now answer the following questions:
(a) Find the mean to determine A’s average number of points scored per game.
Answer: A’s average number of points scored per game can be calculated as follows:
`text(Mean)=text(Sum of each observation)/text(Number of observations)`
`=(14+16+10+10)/(4)`
`=(50)/(4)=12.5`
(b) To find the mean number of points per game for C, would you divide the total points by 3 or by 4? Why?
Answer: To find the mean number of points per game for C, I will divide the total points by 3 because he played in three games only.
(c) B played in all the four games. How would you find the mean?
Answer: Mean score of B can be calculated as follows:
`text(Mean)=text(Sum of each observation)/text(Number of observations)`
`=(0+8+6+4)/(4)`
`=(18)/(4)=4.5`
(d) Who is the best performer?
Answer: A is the best performed with total score of 50. Three players' score in descending order are as follows: 50 > 32 > 18
Question 5: The marks (out of 100) obtained by a group of students in a science test are 85, 76, 90, 85, 39, 48, 56, 95, 81 and 75. Find the:
(a) Highest and the lowest marks obtained by the students.
Answer: These marks can be arranged in ascending order as follows:
39, 48, 56, 75, 76, 81, 85, 85, 90, 95
Highest marks = 95
Lowest marks = 39
(b) Range of the marks obtained.
Answer: Range of marks = Highest marks – Lowest marks `= 95 – 39 = 56`
(c) Mean marks obtained by the group.
Answer: Mean can be calculated as follows:
`text(Mean)=text(Sum of each observation)/text(Number of observations)`
=(39 + 48 + 56 + 75 + 76 + 81 + 85 + 85 + 90 + 95) ÷ 10
`=(730)/(10)=73`
Question 6: The enrolment in a school during six consecutive years was as follows: (1555, 1670, 1750, 2013, 2540, 2820) Find the mean enrolment of the school for this period.
Answer: Mean can be calculated as follows:
`text(Mean)=text(Sum of each observation)/text(Number of observations)`
=(1555 + 1670 + 1750 + 2013 + 2540 + 2820) ÷ 6
`= (12348)/(6) = 2058`
Question 7: The rainfall (in mm) in a city on 7 days of a certain week was recorded as follows:
| Day | Mon | Tue | Wed | Thu | Fri | Sat | Sun |
|---|---|---|---|---|---|---|---|
| Rainfall (in mm) | 0.0 | 12.2 | 2.1 | 0.0 | 20.5 | 5.5 | 1.0 |
(a) Find the range of the rainfall in the above data.
Answer: Range = Highest rainfall – Lowest rainfall
`= 20.5 – 0.0 = 20.5` mm
(b) Find the mean rainfall for the week.
Answer: Mean can be calculated as follows:
`text(Mean)=text(Sum of each observation)/text(Number of observations)`
=(0.0 + 12.2 + 0.0 + 20.5 + 5.5 + 1.0) ÷ 7
`=(39.2)/(7)=5.6` mm
(c) On how many days was the rainfall less than the mean rainfall.
Answer: Five days, i.e. Monday, Wednesday, Thursday, Saturday and Sunday.
Question 8: The heights of 10 girls were measured in cm and the results are as follows: (135, 150, 139, 128, 151, 132, 146, 149, 143, 141)
Answer: Above data can be arranged in ascending order as follows:
128, 132, 135, 139, 141, 143, 146, 149, 150, 151
(a) What is the height of the tallest girl?
Answer: Height of tallest girl = 151 cm
(b) What is the height of the shortest girl?
Answer: Height of shortest girl = 128 cm
(c) What is the range of the data?
Answer: Range of data = Tallest height – Shortest height
= 151 cm – 128 cm = 23 cm
(d) What is the mean height of the girls?
Answer: Mean height can be calculated as follows:
Mean= (Sum of each observation)/(Number of observations)
=(128 + 132 + 135 + 139 + 141 + 143 + 146 + 149 + 150 + 151) ÷ 10
`=(1414)/(10)=141.4`
(e) How many girls have heights more than the mean height?
Answer: Five girls have heights more than the mean height