Statistics
ICSE Class 9 · Mathematics · Chapter 17
What this chapter covers
- Statistics and the collection of data
- Raw, arrayed and grouped data
- Tabulation of raw data using tally marks
- Discrete and continuous variables
- Grouped data: class intervals, limits, class size and frequency
- Class boundaries: converting discontinuous intervals to continuous intervals
- Graphical representation: the frequency polygon
Key definitions
Statistics is the branch of mathematics that deals with the collection, organisation, presentation, analysis and interpretation of numerical facts.
Data are facts or figures, usually numbers, collected for a definite purpose so that some conclusion can be drawn from them.
Primary data are data collected by the investigator personally, directly from the source, for the purpose in hand.
Secondary data are data that were originally collected by someone else, for some other purpose, and are now being used by the investigator.
Raw data are data recorded in the order in which they were collected, without any rearrangement or grouping.
Arrayed data are raw data that have been arranged in ascending or descending order of magnitude.
Grouped data are data arranged into a number of groups, called classes, together with the number of observations that fall in each class.
Key points
- Statistics deals with the collection, organisation, presentation, analysis and interpretation of numerical data.
- Primary data are collected first-hand by the investigator; secondary data are taken from sources collected by others.
- Raw data are in the order collected; arrayed data are arranged in ascending or descending order; grouped data are sorted into classes with frequencies.
- The range is the largest observation minus the smallest.
- The frequency of a value is the number of times it occurs; tally marks are drawn in bunches of five to count frequencies.
- A discrete variable takes separate values, usually by counting; a continuous variable can take any value in a range, usually by measuring.
- In the exclusive form (10–20, 20–30) the upper limit belongs to the next class; in the inclusive form (11–20, 21–30) both limits belong to the class.
- Class size is the difference between the class boundaries, or between consecutive lower limits.
- Class mark = (lower limit + upper limit)/(2); consecutive class marks differ by the class size.
- To make inclusive classes continuous, find the gap between classes, halve it, subtract the half from each lower limit and add it to each upper limit; frequencies and class marks do not change.
- A frequency polygon is drawn by plotting (class mark, frequency), adding an imaginary class with zero frequency at each end, and joining the points with straight line segments.
Questions answered
Define statistics, and distinguish between primary data and secondary data with one example of each.
Statistics is the branch of mathematics that deals with the collection, organisation, presentation, analysis and interpretation of numerical facts. Primary data are collected by the investigator directly for the purpose in hand, such as a student measuring the heights of classmates. Secondary data were collected by someone else for another purpose and are reused, such as rainfall figures taken from a weather bureau's report.
Distinguish between raw data and arrayed data. Arrange 17, 9, 23, 9, 14, 30, 21 as an array and find the range.
Raw data are in the order in which they were collected; arrayed data are the same values arranged in ascending or descending order. The array is 9, 9, 14, 17, 21, 23, 30, and the range is 30 - 9 = 21.
The number of goals scored in 20 football matches was: 1, 2, 0, 3, 1, 2, 2, 1, 0, 4, 1, 2, 3, 1, 2, 0, 1, 2, 1, 3. Prepare a frequency table using tally marks.
Goals 0: three strokes, frequency 3. Goals 1: one gate and two strokes, frequency 7. Goals 2: one gate and one stroke, frequency 6. Goals 3: three strokes, frequency 3. Goals 4: one stroke, frequency 1. Total 3 + 7 + 6 + 3 + 1 = 20.
Why are tally marks drawn in bunches of five?
Bunches of five can be counted quickly in fives, which makes the final count faster and reduces counting errors.
Classify as discrete or continuous: (i) the number of books in a school bag, (ii) the distance from home to school, (iii) the number of runs in an over, (iv) the temperature of a city at noon.
(i) Discrete, as books are counted. (ii) Continuous, as distance is measured and can take any value. (iii) Discrete, as runs are counted. (iv) Continuous, as temperature is measured.
A person's age is usually given in whole years. Explain why age is still a continuous variable.
Age increases continuously with time and can take any value, such as 14 years 3 months 10 days; giving it in completed years only rounds a continuous value.
For the class 35–45 in the distribution 25–35, 35–45, 45–55, write the lower limit, upper limit, class size and class mark. In which class would an observation of exactly 45 be placed?
Lower limit 35, upper limit 45, class size 10, class mark (35 + 45)/(2) = 40. The classes are in exclusive form, so 45 goes into 45–55.
Why is the class size of the class 11–20, in the distribution 1–10, 11–20, 21–30, equal to 10 and not 9?
The class size is the difference between the class boundaries, 20.5 - 10.5 = 10, or between consecutive lower limits, 21 - 11 = 10. Subtracting the limits of an inclusive class ignores the gap between classes and gives the wrong value 9.
Past board exam questions
ICSE 2019 sample paper · 4 marks
Construct a frequency polygon for the following frequency distribution, using a graph sheet. | Marks | 40 – 50 | 50 – 60 | 60 – 70 | 70 – 80 | 80 – 90 | 90 – 100 | |---|---|---|---|---|---|---| | No. of students | 7 | 18 | 26 | 37 | 20 | 6 | Use 2 cm = 10 marks 2 cm = 5 students
Show answer
Class marks: 45, 55, 65, 75, 85, 95 with frequencies 7, 18, 26, 37, 20, 6.
Take marks on the x-axis (2 cm = 10 marks) and number of students on the y-axis (2 cm = 5 students); show a kink on the x-axis since it starts at 30.
Add an imaginary class 30 – 40 and 100 – 110 with zero frequency (class marks 35 and 105).
Plot the points (35, 0), (45, 7), (55, 18), (65, 26), (75, 37), (85, 20), (95, 6), (105, 0) and join them with straight line segments in order. The closed figure is the required frequency polygon.