ICSE Class 10 Maths – Ogive Graph Explained

Ever wondered how you can see the story behind a list of numbers at a glance? That's what an ogive does – it turns raw data into a rising line that tells you how many items are below any given value.

💡 In Simple Words: An ogive is a line graph that shows cumulative frequency – the running total of data points up to each class limit. It helps you answer questions like “how many students scored 70 or less?” in a flash.

What is an Ogive?

In statistics, an ogive (pronounced oh‑jive) is a graph that plots cumulative frequency against the upper class boundaries. Cumulative frequency means you keep adding the frequencies as you move from the lowest class to the highest, just like stacking books one on top of another.

When Do You Use an Ogive?

• To find the number of observations below a certain value.
• To compare two data sets quickly.
• In exam questions that ask for “less than” or “greater than” counts.

Steps to Draw an Ogive (Simple Flow)

Step‑by‑Step Explanation

1. Collect data: Gather the raw numbers you need to analyse. For example, test scores of a class.
2. Arrange in class intervals: Group the numbers into ranges (like 0‑9, 10‑19, …). These ranges are called class intervals.
3. Calculate cumulative frequency: Start with the first interval’s frequency, then keep adding the next interval’s frequency to the total. Write each total next to the upper limit of the interval.
4. Mark points on graph: On the horizontal axis (x‑axis) put the upper class boundaries. On the vertical axis (y‑axis) put the cumulative frequencies. Plot a point for each pair.
5. Connect the points: Use a smooth line or straight segments to join the points from left to right. That line is the ogive.
6. Read values: To find how many observations are below a certain value, drop a vertical line from that value to the ogive, then read the corresponding frequency on the y‑axis.

Worked Example

Suppose 30 students took a math test and their scores are grouped as follows:

First, find the cumulative frequency (CF):

• 0‑9 : CF = 2
• 10‑19 : CF = 2+3 = 5
• 20‑29 : CF = 5+5 = 10
• 30‑39 : CF = 10+8 = 18
• 40‑49 : CF = 18+7 = 25
• 50‑59 : CF = 25+5 = 30

Now plot the points (upper boundary, CF): (9,2), (19,5), (29,10), (39,18), (49,25), (59,30). Connect them and you have the ogive. If a question asks, “How many students scored 35 or less?” you locate 35 on the x‑axis, draw a line up to the curve (it will be between the points for 29 and 39), and read the approximate frequency – about 15 students.

Ogive vs. Other Graphs

Quick Tips for the Exam

• Always start the ogive at the lowest class boundary with a frequency of zero.
• Use the upper class boundary, not the midpoint, for the x‑coordinate.
• Label axes clearly: “Upper Class Boundary” on x‑axis, “Cumulative Frequency” on y‑axis.
• When reading a value, draw a vertical line first, then a horizontal line to the axis – this avoids mistakes.
• Check that the final point reaches the total number of observations (N).

📝 Likely Exam Questions

1. Question: The following table shows the ages of 40 students. Draw the ogive and find how many students are 15 years old or younger.
   Answer: Compute cumulative frequencies, plot points at upper age limits, connect them. The point at 15 (or the nearest upper limit) gives a cumulative frequency of 22, so 22 students are ≤15 years.
2. Question: Explain why an ogive starts at zero on the y‑axis.
   Answer: Because before the first class interval there are no observations, so the cumulative count is zero.
3. Question: A class scored 0‑20, 21‑40, 41‑60, 61‑80, 81‑100 with frequencies 4, 7, 12, 9, 8. Sketch the ogive and state how many scored above 70.
   Answer: After drawing, read the cumulative frequency at the upper boundary 70 (which lies in the 61‑80 class). Suppose the cumulative frequency at 70 is 30; total students = 40, so above 70 = 40‑30 = 10.
4. Question: Compare an ogive with a histogram in two sentences.
   Answer: An ogive shows cumulative totals, helping answer “how many ≤ X?” questions, while a histogram displays the frequency of each class, revealing the shape of the distribution.
5. Question: If the ogive for a data set passes through (50, 20) and (70, 35), estimate the number of observations between 50 and 70.
   Answer: Subtract the cumulative frequencies: 35‑20 = 15 observations lie in that interval.