Tenth Grade Mathematics: Cumulative Frequency Distributions and Ogives (Less Than / More Than)
While tabular cumulative frequencies provide numerical totals, graphical curves offer immediate visual insights into the percentiles, medians, and distribution spreads of large populations. A cumulative frequency curve—traditionally known in statistics as an Ogive (pronounced o-jive)—plots cumulative frequencies against class boundaries. By graphing "Less Than" and "More Than" ogives, statisticians can determine the median of a grouped dataset directly from the intersection of two smooth curves without performing algebraic calculations.
Constructing "Less Than" Cumulative Frequency Distributions
The "Less Than" ogive tracks how many observations fall below each upper class limit:
Protocol for "Less Than" Distribution:
1. Use the UPPER LIMIT of each class interval as the x-coordinate.
2. Accumulate frequencies progressively from the top class downwards.
3. Plot points (Upper Limit, Corresponding Cumulative Frequency).
4. Connect the plotted points with a freehand smooth continuous curve.
Example Tabulation:
Class Interval Frequency Cumulative Frequency (cf) Plotted Point (x, y)
0 - 10 5 5 (10, 5)
10 - 20 8 5 + 8 = 13 (20, 13)
20 - 30 12 13 + 12 = 25 (30, 25)
30 - 40 15 25 + 15 = 40 (40, 40)
40 - 50 10 40 + 10 = 50 (50, 50)
Because cumulative frequencies increase monotonically, a "Less Than" ogive is an ever-rising, S-shaped curve (an ascending ogive).
Constructing "More Than" Cumulative Frequency Distributions
The "More Than" ogive tracks how many observations fall above or equal to each lower class limit:
Protocol for "More Than" Distribution:
1. Use the LOWER LIMIT of each class interval as the x-coordinate.
2. Accumulate frequencies from the bottom class upwards (or subtract progressively
from the total frequency N).
3. Plot points (Lower Limit, Corresponding Cumulative Frequency).
4. Connect the plotted points with a freehand smooth curve.
Example Tabulation (Total N = 50):
Class Interval Frequency "More Than" Condition Cumulative (cf) Plotted Point
0 - 10 5 More than or equal to 0 50 (0, 50)
10 - 20 8 More than or equal to 10 50 - 5 = 45 (10, 45)
20 - 30 12 More than or equal to 20 45 - 8 = 37 (20, 37)
30 - 40 15 More than or equal to 30 37 - 12 = 25 (30, 25)
40 - 50 10 More than or equal to 40 25 - 15 = 10 (40, 10)
A "More Than" ogive is an ever-falling, descending curve.
Determining the Median Graphically
There are two distinct graphical techniques to locate the median using ogives:
Method 1: Using a SINGLE "Less Than" Ogive
1. Draw the "Less Than" ogive.
2. Locate the value N / 2 on the vertical y-axis.
3. From N / 2, draw a horizontal line parallel to the x-axis to intersect the ogive
at point P.
4. From P, drop a vertical perpendicular line to meet the horizontal x-axis at M.
5. The x-coordinate of point M is the MEDIAN!
Method 2: Using the INTERSECTION of Both Ogives
1. Plot both the "Less Than" ogive and the "More Than" ogive on the SAME coordinate axes.
2. The two curves will intersect at exactly one unique point P.
3. From the intersection point P, drop a perpendicular to the horizontal x-axis.
4. The x-coordinate where the perpendicular hits the x-axis is the MEDIAN!
(The y-coordinate of the intersection point is always identically N / 2).
Visual Scheme of Ogive Intersection:
Cumulative Frequency
^
N + "More Than" Ogive "Less Than" Ogive
| \ /
| \ /
N / 2 + - - \ - - - - - - - - - - / - - Point of Intersection P (Median, N/2)
| \ P /
| \ * /
| \ / \ /
| \ / \ /
+----------\/-----+----+-----------> Value
0 | Upper Limits
Median
Determining Quartiles from an Ogive
In addition to the median (which is the second quartile Q2), ogives allow graphical determination of the lower quartile Q1 and upper quartile Q3: Lower Quartile Q1: Drop perpendicular from y = N / 4. Median Q2: Drop perpendicular from y = 2N / 4 = N / 2. Upper Quartile Q3: Drop perpendicular from y = 3N / 4. Interquartile Range = Q3 - Q1.
Chapter Exercises
Exercise 1: The following distribution gives the state-wise literacy rates of 35 cities: 45-55% (3 cities), 55-65% (10 cities), 65-75% (11 cities), 75-85% (8 cities), 85-95% (3 cities). Convert the distribution to a "Less Than" type cumulative frequency distribution and state the coordinates to plot its ogive.
Exercise 2: For the data in Exercise 1, construct the "More Than" cumulative frequency table and write down the coordinates for plotting the "More Than" ogive.
Exercise 3: Explain why the y-coordinate of the point of intersection of the "Less Than" and "More Than" ogives for any dataset must be equal to half the total frequency (N / 2).
Solutions and Explanations
Solution 1: We construct the "Less Than" cumulative frequency table using upper limits: Less than 55% (cf = 3, point (55, 3)); Less than 65% (cf = 3 + 10 = 13, point (65, 13)); Less than 75% (cf = 13 + 11 = 24, point (75, 24)); Less than 85% (cf = 24 + 8 = 32, point (85, 32)); Less than 95% (cf = 32 + 3 = 35, point (95, 35)). The coordinates to plot the "Less Than" ogive on a graph are (55, 3), (65, 13), (75, 24), (85, 32), and (95, 35).
Solution 2: For the "More Than" cumulative frequency distribution, we use lower limits and accumulate from the total frequency N = 35: More than or equal to 45% (cf = 35, point (45, 35)); More than or equal to 55% (cf = 35 - 3 = 32, point (55, 32)); More than or equal to 65% (cf = 32 - 10 = 22, point (65, 22)); More than or equal to 75% (cf = 22 - 11 = 11, point (75, 11)); More than or equal to 85% (cf = 11 - 8 = 3, point (85, 3)). The coordinates to plot the "More Than" ogive are (45, 35), (55, 32), (65, 22), (75, 11), and (85, 3).
Solution 3: Let any value along the horizontal axis be x. By definition, the "Less Than" cumulative frequency at x represents the number of observations strictly less than x: cf_less = Count(Value < x). The "More Than" cumulative frequency represents the number of observations greater than or equal to x: cf_more = Count(Value >= x). Because every observation in the dataset of size N must be either less than x or greater than or equal to x, their sum at any point is identically the total population: cf_less + cf_more = N. At the unique point of intersection of the two curves, their vertical heights are identical: cf_less = cf_more. Substituting this into the sum equation gives: cf_intersect + cf_intersect = N ==> 2 * cf_intersect = N ==> cf_intersect = N / 2. Therefore, the y-coordinate at the intersection point is always strictly N / 2. By definition of the median, the value on the x-axis that corresponds to N / 2 cumulative observations is the median.