Classification and Presentation of Data

You drew bar diagrams in earlier grades. Here is one: the number of students in each grade, from Grade 5 to Grade 10, of Shree Janabikash Secondary School.

A bar diagram works when the things being counted come in separate kinds: Grade 7, Grade 8. But marks, heights, wages and ages run on a scale with no gaps. This unit shows how to count such data in tables and how to draw it.

Activity 1: forty marks in two tables

These are three ways of holding the same data, and each has a name.

• The plain list, one value for each person, in any order, is an individual series (also called raw data).
• Table I, each value with the number of times it occurs, is a discrete series. A table like I or II is a frequency table (or frequency distribution), and a series is also called a distribution: a discrete distribution, a continuous distribution. The number of times a value occurs is its frequency, f, and the total of the frequencies is N, the number of values.
• Table II, the values counted in groups, is a continuous series. Each group, like 40–50, is a class interval (or class). Its smaller number, 40, is its lower limit, and its bigger number, 50, its upper limit; the difference, 50 − 40 = 10, is the class width (or size of the class).

To count, go through the list once and make a tally mark for each value in its row: four strokes, then a fifth across them, so the marks fall into bundles of five that are quick to count (|||| |||| || is 5 + 5 + 2 = 12). Then cross the value off the list, so none is counted twice.

Worked example: a discrete series

Cumulative frequency

A cumulative frequency (cf) is a running total: the frequencies added one by one, in order. Each one answers “how many so far?”. Put the values in order first (increasing or decreasing), or the running total means nothing. Two checks never fail: the first cf is the first frequency, and the last cf is N.

Less than and more than

So a continuous series has two cumulative frequency tables:

• the less-than table, with each class’s upper limit: it starts with the first class’s frequency and grows to N;
• the more-than table, with each class’s lower limit: it starts at N and shrinks to the last class’s frequency.

Question 1

The marks (out of 10) of 30 Grade 9 students in a Science unit test:
6, 8, 10, 6, 2, 8, 4, 6, 8, 2, 4, 6, 8, 6, 8, 6, 10, 2, 4, 6, 8, 4, 2, 4, 8, 6, 4, 6, 10, 6

Question 2

Question 3

Use the continuous series of Question 2.

Question 4

Question 5

The time 1,025 students spend studying Mathematics each week:

Hours	2–6	6–10	10–14	14–18	18–22
Students	100	250	325	200	150

A table holds the numbers; a graph shows their shape at a glance. A continuous series has three graphs, and the whole difference between them is what goes on the x-axis:

Graph	x-axis	y-axis	Joined by
Histogram	the class intervals	frequency	touching rectangles
Frequency polygon	each class’s mid-value	frequency	straight lines (a ruler)
Ogive	upper limits (less than), lower limits (more than)	cumulative frequency	a smooth curve, freehand

A histogram is a graph of a continuous series made of rectangles side by side: each one stands on its class interval on the x-axis, and its height is the class’s frequency. So its width is the class width and its area is in proportion to the frequency. A bar diagram is for discrete data (separate kinds or values); a histogram is for continuous data.

Steps to draw a histogram

• a.Draw the x-axis and the y-axis on graph paper.
• b.Mark the class limits on the x-axis and the frequencies on the y-axis, each with a suitable scale.
• c.On each class interval draw a rectangle as high as its frequency, the rectangles joined edge to edge.

A frequency polygon is the line graph made by plotting each class’s mid-value on the x-axis against its frequency on the y-axis, and joining the points one after another with a ruler (straight lines). The mid-value of a class is

$$\text{mid-value} = \frac{\text{lower limit} + \text{upper limit}}{2}$$

Steps to draw a frequency polygon

• a.Draw the x-axis and the y-axis on graph paper.
• b.Find each class’s mid-value. Mark the mid-values on the x-axis and the frequencies on the y-axis, each with a suitable scale.
• c.Plot the points (mid-value, frequency) and join them in order with a ruler.
• d.Add one empty class of the same width before the first class and one after the last. Plot their mid-values with frequency 0, and join them to the ends. The polygon is closed on the x-axis.

Why add the empty classes? Watch what the polygon does to the histogram’s area.

A cumulative frequency curve, or ogive, is the curve through the points (class limit, cumulative frequency), drawn freehand, without a ruler, as one smooth curve.

• The less-than ogive plots each class’s upper limit against its less-than cumulative frequency. It rises.
• The more-than ogive plots each class’s lower limit against its more-than cumulative frequency. It falls.

Steps to draw a less-than ogive

• a.Make the less-than cumulative frequency table.
• b.Draw the x-axis and the y-axis on graph paper.
• c.Mark the upper limits on the x-axis with a suitable scale.
• d.Mark the cumulative frequencies on the y-axis with a suitable scale.
• e.Plot the points (upper limit, cumulative frequency).
• f.Join them freehand with a smooth curve.

Steps to draw a more-than ogive: the same, with the more-than table and the points (lower limit, cumulative frequency).

The opening asked where the biggest group of the 40 Grade 8 students scored, and how many scored below 60.

The biggest group scored in the 40s (12 students), and 16 students scored below 60.

The continuous series of Activity 1 (table II) gives the first at once. Its less-than table gives the second, and more:

Marks	Students
Less than 50	12
Less than 60	16
Less than 70	23
Less than 80	31
Less than 90	36
Less than 100	40

So 40 − 31 = 9 students scored 80 or more. And the class has two humps, one in the 40s and one in the 60s and 70s: a teacher might wonder whether the 12 students in the 40s needed more help than the rest. The list of 40 numbers hid all of this; the table and the histogram show it.

Grade 9’s final exam is set to CDC’s specification grid for Compulsory Mathematics, Grades 9–10. There, Statistics and Probability is 11 marks in two questions. In the grid’s model paper, question 14 gives the marks of 20 students as a discrete series:

Part	Level	Marks	What it asks
a.	knowledge	1	in which series the data is presented
b.	understanding	1	a cumulative frequency table
c.	application	2	the median (Unit 18)
d.	higher ability	1	the mean of the marks above the median (Unit 18)

Parts a and b are this unit. Here is one of our own in the same shape, with c and d on what this unit teaches: the ogive.

Question 1

Draw a histogram of each.

Question 2

Draw a frequency polygon of each.

Question 3

Draw the less-than and the more-than ogive of each.

Question 4

Question 5

Question 6

Question 7

A survey measured the protein in 40 different foods (grams):
23, 30, 20, 27, 44, 26, 35, 20, 29, 29, 25, 15, 18, 27, 19, 22, 12, 26, 34, 15, 27, 35, 26, 43, 35, 14, 24, 12, 23, 31, 40, 35, 38, 57, 22, 42, 24, 21, 27, 33