Lesson 3&4
Class:SS2 & SS3
Subject: Mathematics
Topic: Cumulative Frequency Graph (Ogive)
It has been stated earlier in our previous lesson that the sum of all frequencies from the first class to a particular class is referred to as the cumulative frequency of that class. The graph of cumulative frequency distribution is a cumulative frequency curve which is also called the ogive.
The cumulative frequency graph has an elongated S – shape curve. It is obtained by plotting cumulative frequency on the y-axis against the corresponding upper class boundary on x-axis.
The cumulative frequency curve ( ogive) can be used to find the median, quartile, percentiles, deciles and interquartile range etc.
Determination of median, quartiLEs, deciles and percentiles from cumulative frequency graph.
- 1. Median: This is the score that corresponds to the middle score half way up the cumulative frequency axis. It is also called the second quartile Q2. It is given by 1/2*N score on the cumulative frequency axis, where N is the total frequency.
The median can be determined as follows:
- Determine half of the total frequency. That is 1/2*N
- Locate this point on the cumulative frequency axis on the graph.
- Through this located point on the cumulative frequency axis, draw a line parallel to the x-axis to meet the curve.
- From this point of intersection, draw a perpendicular line to the x- axis.
- Locate and determine the value of the point at which this vertical line meets the x- axis. This gives the median.
2.Quartile: The quartile is an average that divides the whole distribution (data) into four equal parts namely:
First or lower quartile (Q1): Is one- quarter up the cumulative frequency axis of the graph. Assuming the total frequency of a given distribution is N, then the lower quartile Q1 is given by 1/4*N score.
The lower quartile is determine from cumulative frequency graph as follows:
- Determine one-quarter of the total frequency. That is 1/4*N
Repeat steps b to e as in (1) above to obtain the lower quartile.
upper quartile (Q3) is the third quartile of the distribution. As the name implies, it is three- quarter up the cumulative frequency axis of the graph. The upper quartile Q3 is given by 3/4*N score on the graph. The upper quartile is obtained from cumulative frequency graph as follows:
- Determine three-quarter of the total frequency. That is 3/4*N
Repeat steps b to e as in (1) above to obtain the upper quartile Q3.
Interquartile range is the difference between the upper quartile and lower quartile values written as Q= Q3 – Q1.
semi-Interquartile range is the Interquartile range divided by two. That is
(Q3 -Q1)/2.
- 3. Decile: Decile divides the data into ten equal parts. For instance, the 7th decile is 7/10*N. It is obtained from cumulative frequency graph as follows:
- Determine 7/10*N on cumulative frequency axis
Repeat steps b to e as in (1) above to obtain the 7th decile.
- 4. Percentile: Just like the quartile and decile, the percentile divides the data into hundred equal parts. For instance, the 30th percentile is 30/100*N. It is obtained from cumulative frequency graph as follows:
- Determine 30/100*N on cumulative frequency axis
Repeat steps b to e as in (1) above to obtain the 30th percentile
Example 1: The following shows the distribution of scores obtained by SS2 students in mathematics mock exam.
| Class interval | 21-30 | 31-40 | 41-50 | 51-60 | 61-70 | 71-80 | 81-90 | 90-100 |
| Frequency | 2 | 6 | 7 | 10 | 11 | 9 | 4 | 1 |
- Construct a cumulative frequency table for the distribution.
- Draw a cumulative frequency curve for the distribution.
- From your graph, estimate the:
- median
- interquartile range
iii. pass mark, if 70% of the students passed.
. Solution
a.
| Class interval | Frequency | Class boundary | Cumulative frequency |
| 21-30 | 2 | 20.5-30.5 | 2 |
| 31-40 | 6 | 30.5-40.5 | 8 |
| 41-50 | 7 | 40.5-50.5 | 15 |
| 51-60 | 10 | 50.5-60.5 | 25 |
| 61-70 | 11 | 60.5-70.5 | 36 |
| 71-80 | 9 | 70.5-80.5 | 45 |
| 81-90 | 4 | 80.5-90.5 | 49 |
| 91-100 | 1 | 90.5-100.5 | 50 |
- ci.
Given N=50,
Recall that, median= 1/2*N. That is 1/2*50=25
From the graph, the median =63 (That is 25 on the Y-axis corresponds to 63 on the x-axis on the graph)
cii.
Q3 =3/4*N, That is 3/4*50=37.5
From the graph, Q3=73
Q1 =1/4*N, That is 1/4*50=12.5
From the graph, Q1=48.
Hence the interquartile range(Q) =Q3-Q1
Q=75-48
=25
ciii.
If 70% of the students passed the exam, it means 30% failed. We therefore, need to find only 30th percentile on the cumulative frequency graph; above this point we have the 70% needed. That is
30th percentile (P30) =30/100*N
P30=30/100*50=15
From the graph, P30=50.5
Therefore the pass mark is 50.5
Assignment
The table below shows the frequency distribution of marks scored by 120 students in Mathematics test.
| Marks% | 10-19 | 20-29 | 30-39 | 40-49 | 50-59 | 60-69 | 70-79 | 80-89 |
| Frequency | 23 | 19 | 6 | 25 | 15 | 11 | 12 | 9 |
- Prepare a cumulative frequency table and use it to draw the ogive for the data.
- Use your graph to estimate the:
- lower quartile
- 70th percentile
- Find the pass mark if 35 candidates passed the exam. (Hint: Get the pass mark from the y-axis)
DOWNLOAD DOCUMENT HERE: SS2 Mathematics (lesson 3&4)
N/B: Always refers to your Mathematics textbooks for further studies.
Submit the assignment on the 12th June, 2020 via whatsApp @07035773326 or in hardcopy at the security post in the school.