Statistics • Topic 3 of 3

Histograms, Frequency Polygons, and Ogives

What are histograms, frequency polygons, and ogives? These are graphical representations of frequency distributions that help visualize data patterns.

1. Histogram: A histogram is a bar graph where:

  • Bars are drawn without gaps (touching each other)
  • Width of each bar = class width
  • Height of each bar = frequency of that class
  • The x-axis shows class boundaries, y-axis shows frequency

2. Frequency Polygon: A frequency polygon is a line graph created by:

  • Plotting points at class midpoints with heights equal to frequencies
  • Joining these points with straight lines
  • Adding two extra points at the ends (with zero frequency) to close the polygon

3. Ogive (Cumulative Frequency Curve): An ogive is a smooth curve drawn using cumulative frequencies:

  • Less than ogive: Plot upper limits vs less than cumulative frequency, then join with a smooth curve
  • Greater than ogive: Plot lower limits vs greater than cumulative frequency, then join with a smooth curve

Finding median from ogives: The x-coordinate of the intersection of the less than and greater than ogives gives the median.

┌─────────────────────────────────────────────────────────────┐
│       HISTOGRAM, FREQUENCY POLYGON, AND OGIVE - PLOTS        │
└─────────────────────────────────────────────────────────────┘

SAMPLE DATA: Class: 0-10,10-20,20-30,30-40,40-50; Freq: 5,8,12,7,3

HISTOGRAM (Bars touching):

    Frequency
       12 ┤        ┌─────┐
       10 ┤        │     │
        8 ┤   ┌────┤     │
        6 ┤   │    │     │
        4 ┤   │    │     │   ┌─┐
        2 ┤   │    │     │   │ │
        0 └───┴────┴─────┴───┴─┴──► Class
            0-10 10-20 20-30 30-40 40-50


FREQUENCY POLYGON (Points at midpoints: 5,15,25,35,45)

    Frequency
       12 ┤        ●
       10 ┤       / \
        8 ┤      /   \
        6 ┤     /     \
        4 ┤    /       \
        2 ┤   /         \
        0 ┼──●───────────●──●──►
           -5  5  15 25 35 45 55
              (Extra points at 0 and 50 to close polygon)


OGIVE (Less than cumulative curve):

    Cumulative
    Frequency
       35 ┤                              ●
       30 ┤                        ●─────┘
       25 ┤                  ●─────┘
       20 ┤            ●─────┘
       15 ┤      ●─────┘
       10 ┤●─────┘
        5 ┤
        0 └─────┴─────┴─────┴─────┴─────► Upper Limit
           10    20    30    40    50


OGIVE (Both less than and greater than on same axes):

    Cumulative
    Frequency
       35 ┤  ●                              
       30 ┤  │  ○                          
       25 ┤  │  │  ●                      
       20 ┤  │  │  │  ○                  
       15 ┤  │  │  │  │  ●              
       10 ┤  │  │  │  │  │  ○          
        5 ┤  │  │  │  │  │  │  ●      
        0 └──┼──┼──┼──┼──┼──┼──►
            10 20 30 40 50 60
           Less than (●)  Greater than (○)
           
           INTERSECTION point gives MEDIAN!
Solved Examples
1
Worked Example
Draw a histogram for the data: Class 0-10 (f=4), 10-20 (f=6), 20-30 (f=10), 30-40 (f=5).
Solution
  1. Step 1: Identify class boundaries: 0-10, 10-20, 20-30, 30-40
  2. Step 2: Bars have width 10, heights = frequencies
  3. Step 3: Bars touch each other (no gaps)

Answer: Histogram drawn with bars at heights 4,6,10,5

2
Worked Example
From the following less than ogive points, estimate the median: (10,4), (20,10), (30,22), (40,30), (50,35)
Solution
  1. Step 1: Total frequency N = 35
  2. Step 2: N/2 = 17.5
  3. Step 3: On ogive, find x-coordinate where cumulative frequency = 17.5
  4. Step 4: 17.5 lies between (20,10) and (30,22)
  5. Step 5: Interpolate: 20 + [(17.5-10)/(22-10)] × 10 = 20 + (7.5/12)×10 = 20 + 6.25 = 26.25

Answer: Median ≈ 26.25

3
Worked Example
Using the data in Example 1, find the number of students scoring between 15 and 35 marks (using frequency polygon).
Solution
  1. Step 1: Midpoints: 5, 15, 25, 35; frequencies: 4, 6, 10, 5
  2. Step 2: Frequencies for classes containing 15-35: class 10-20 (15 is midpoint), 20-30, 30-40 (35 is midpoint)
  3. Step 3: These classes cover 15 to 35 completely
  4. Step 4: Total frequency = 6 + 10 + 5 = 21 students

Answer: 21 students (within 15-35 marks)

Key Points

  • Histogram: Touching bars for continuous data; area of bar represents frequency
  • Frequency polygon: Points at midpoints joined by lines; starts and ends at zero frequency
  • Less than ogive: Upper limits vs less than cumulative frequency
  • Greater than ogive: Lower limits vs greater than cumulative frequency
  • Median can be found at intersection of the two ogives
  • Ogives are useful for finding percentiles and quartiles
  • Histograms show distribution shape (symmetric, skewed left, skewed right)
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.A histogram represents grouped data using:
Explanation: Adjacent bars.
Q2.An ogive is a:
Explanation: Cumulative frequency curve.
Q3.A frequency polygon is drawn by joining the:
Explanation: Midpoints of class tops.
Q4.In a histogram, the bars are drawn:
Explanation: No gaps (continuous data).
Practice more MCQs → 📝 Assignment · 35 marks