**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**.