CBSE Class 10 Maths: Statistics Exercise 14.4 - Ogives and Graphical Median

Understanding Ogives (Cumulative Frequency Curves)

An Ogive (pronounced oh-jive or o-jeev) is a smooth curve or a frequency polygon that represents the cumulative frequency distribution of a set of data. Unlike a histogram that shows individual class frequencies, an ogive highlights the running total of frequencies up to a certain point. This graphical representation is incredibly useful for several reasons, including:

1. Visualizing Cumulative Frequencies: It provides a clear picture of how many observations fall below or above a certain value.
2. Estimating Median: One of its most significant applications is the graphical determination of the median, which is the middle value of a data set when arranged in order.
3. Comparing Distributions: By plotting multiple ogives on the same graph, one can easily compare different data sets.

There are two main types of Ogives:

• Less Than Ogive: This curve shows the number of observations less than the upper class boundary of each class interval. To draw this, we plot the upper class limits on the x-axis and the corresponding 'less than' cumulative frequencies on the y-axis.
• More Than Ogive: This curve illustrates the number of observations greater than or equal to the lower class boundary of each class interval. For this, we plot the lower class limits on the x-axis and the corresponding 'more than' cumulative frequencies on the y-axis.

The intersection point of the 'less than' and 'more than' Ogives provides the median of the data set. Understanding this concept is key to solving problems in Exercise 14.4 effectively.

Key Terms in Cumulative Frequency and Ogives

Cumulative Frequency (CF)

The cumulative frequency of a class is the sum of the frequencies of that class and all classes below it (for 'less than' type) or above it (for 'more than' type).

Class Limit

The minimum and maximum values within a class interval. The lower class limit is the smallest value, and the upper class limit is the largest value in a class.

Less Than Ogive

A cumulative frequency curve where points are plotted using the upper class limits and their corresponding 'less than' cumulative frequencies. The curve is typically upward sloping.

More Than Ogive

A cumulative frequency curve where points are plotted using the lower class limits and their corresponding 'more than' cumulative frequencies. The curve is typically downward sloping.

Median Graphically

The value on the x-axis corresponding to the point where the 'less than' and 'more than' Ogives intersect, or by finding the value corresponding to N/2 on the y-axis from a single Ogive.

Steps to Construct Ogives and Find the Median Graphically

1. Step 1: Prepare the Frequency Distribution Table — Start with the given frequency distribution. Ensure the class intervals are continuous. If not, make them continuous by adjusting the limits (subtract 0.5 from lower limit, add 0.5 to upper limit of each class, if discrete data is given).
2. Step 2: Calculate Cumulative Frequencies — For a 'less than' ogive, calculate 'less than' cumulative frequencies by adding frequencies cumulatively from the bottom. For a 'more than' ogive, calculate 'more than' cumulative frequencies by subtracting frequencies cumulatively from the total, starting from the last class, or summing from the top downwards but associating with lower class limits.
3. Step 3: Choose Appropriate Axes — Draw horizontal (x-axis) and vertical (y-axis) axes. Label the x-axis with the 'class limits' (upper for 'less than', lower for 'more than') and the y-axis with 'cumulative frequency'.
4. Step 4: Plot Points for the Ogive — For a 'less than' ogive, plot points (upper class limit, less than cumulative frequency). For a 'more than' ogive, plot points (lower class limit, more than cumulative frequency). Include a starting point (0,0) or (lower limit of first class, 0) for less than ogive, and a final point (upper limit of last class, 0) for more than ogive to ensure smooth curves.
5. Step 5: Draw the Ogive Curve — Connect the plotted points with a smooth, freehand curve. Ensure the curve is smooth and does not look like a series of straight lines.
6. Step 6: Find the Median Graphically — To find the median, calculate N/2, where N is the total frequency. Locate N/2 on the y-axis. Draw a horizontal line from this point to intersect the Ogive. From the intersection point, draw a vertical line down to the x-axis. The value on the x-axis where this vertical line meets is the median. If both ogives are drawn, the x-coordinate of their intersection point gives the median.

Worked Examples: Constructing Ogives and Finding Median

• Example 1: Construct a 'less than' Ogive and find the median. The following table gives the production yield per hectare of wheat of 100 farms of a village: | Production Yield (in kg) | Number of Farms | |---|---| | 50-55 | 2 | | 55-60 | 8 | | 60-65 | 12 | | 65-70 | 24 | | 70-75 | 38 | | 75-80 | 16 | Solution: Step 1: Create 'less than' cumulative frequency table. | Production Yield (Upper Class Limit) | Number of Farms (f) | Cumulative Frequency (cf) | |---|---|---| | Less than 55 | 2 | 2 | | Less than 60 | 8 | 2+8=10 | | Less than 65 | 12 | 10+12=22 | | Less than 70 | 24 | 22+24=46 | | Less than 75 | 38 | 46+38=84 | | Less than 80 | 16 | 84+16=100 | Step 2: Plot the points. Plot (55, 2), (60, 10), (65, 22), (70, 46), (75, 84), (80, 100). Also, plot (50, 0) for continuity. Step 3: Draw a smooth curve through these points. This is the 'less than' Ogive. Step 4: Find the median. Total number of farms (N) = 100. So, N/2 = 100/2 = 50. Locate 50 on the y-axis. Draw a horizontal line from (0, 50) to intersect the Ogive. From the intersection point, draw a vertical line down to the x-axis. The value on the x-axis will be approximately 70.5. So, the median production yield is 70.5 kg/hectare.

Exam Tips for Ogives and Median

1. Choose the Right Class Limits for Plotting:

• For a 'less than' ogive, always use the upper class limits on the x-axis.
• For a 'more than' ogive, always use the lower class limits on the x-axis.
• Make sure your class intervals are continuous. If they are not (e.g., 0-9, 10-19), adjust them to make them continuous (e.g., 0-9.5, 9.5-19.5).

2. Accurate Cumulative Frequency Calculation:

• Double-check your cumulative frequency calculations. A small error here will lead to an incorrect ogive and median.
• For 'less than' CF, the last CF should equal the total frequency (N).
• For 'more than' CF, the first CF (corresponding to the lowest class limit) should equal N, and the last CF should be 0.

3. Label Axes Clearly:

• Always label your x-axis (e.g., 'Upper Class Limits', 'Production Yield') and y-axis ('Cumulative Frequency'). Include units if applicable.

4. Smooth Curve, Not Straight Lines:

• Connect the plotted points with a smooth, freehand curve. Do not use a ruler to draw straight line segments between points, as this defeats the purpose of a curve.

5. Precision in Median Finding:

• When finding the median graphically, use a ruler to draw precise lines from N/2 on the y-axis to the curve and then to the x-axis. Read the value on the x-axis carefully. Practice reading values accurately from graphs.

Practice Questions with Solutions

• Q: The following distribution gives the daily income of 50 workers of a factory. | Daily Income (₹) | Number of Workers | |---|---| | 100-120 | 12 | | 120-140 | 14 | | 140-160 | 8 | | 160-180 | 6 | | 180-200 | 10 | Draw a 'less than' type Ogive for the given data and find the median daily income. A: Step 1: Create 'less than' cumulative frequency table. | Daily Income (Upper Limit) | Number of Workers | Cumulative Frequency | |---|---|---| | < 120 | 12 | 12 | | < 140 | 14 | 12+14=26 | | < 160 | 8 | 26+8=34 | | < 180 | 6 | 34+6=40 | | < 200 | 10 | 40+10=50 | Step 2: Plot the points (120, 12), (140, 26), (160, 34), (180, 40), (200, 50). Start with (100, 0). Step 3: Draw a smooth curve. Total frequency N = 50, so N/2 = 25. Step 4: Locate 25 on the y-axis. Draw a horizontal line to the Ogive and then a vertical line to the x-axis. The value on the x-axis will be approximately 139.5. Final answer: The median daily income is approximately ₹139.5.
• Q: During the medical check-up of 35 students of a class, their weights were recorded as follows: | Weight (in kg) | Number of Students | |---|---| | Less than 38 | 0 | | Less than 40 | 3 | | Less than 42 | 5 | | Less than 44 | 9 | | Less than 46 | 14 | | Less than 48 | 28 | | Less than 50 | 32 | | Less than 52 | 35 | Draw a 'less than' type Ogive for the given data. Find the median weight from the graph. A: Step 1: The data is already in 'less than' cumulative frequency form. Step 2: Plot the points (38, 0), (40, 3), (42, 5), (44, 9), (46, 14), (48, 28), (50, 32), (52, 35). Step 3: Draw a smooth curve connecting these points. Total frequency N = 35, so N/2 = 17.5. Step 4: Locate 17.5 on the y-axis. Draw a horizontal line to the Ogive and then a vertical line to the x-axis. The value on the x-axis will be approximately 46.5. Final answer: The median weight is approximately 46.5 kg.
• Q: The following table gives the life times of 400 neon lamps: | Life time (in hours) | Number of lamps | |---|---| | 1500-2000 | 14 | | 2000-2500 | 56 | | 2500-3000 | 60 | | 3000-3500 | 86 | | 3500-4000 | 74 | | 4000-4500 | 62 | | 4500-5000 | 48 | Draw a 'more than' type Ogive for the given data. A: Step 1: Create 'more than' cumulative frequency table. Total lamps N = 400. | Life Time (Lower Limit) | Number of lamps | 'More Than' Cumulative Frequency | |---|---|---| | More than or equal to 1500 | 400 | 400 | | More than or equal to 2000 | 400-14=386 | 386 | | More than or equal to 2500 | 386-56=330 | 330 | | More than or equal to 3000 | 330-60=270 | 270 | | More than or equal to 3500 | 270-86=184 | 184 | | More than or equal to 4000 | 184-74=110 | 110 | | More than or equal to 4500 | 110-62=48 | 48 | Step 2: Plot the points (1500, 400), (2000, 386), (2500, 330), (3000, 270), (3500, 184), (4000, 110), (4500, 48), (5000, 0). Step 3: Draw a smooth curve connecting these points. This is the 'more than' Ogive. Final answer: The graph plotted with the above points forms the 'more than' type Ogive.
• Q: The following table gives the daily wages of 100 workers. Construct both 'less than' and 'more than' type Ogives on the same graph and find the median. | Daily Wages (₹) | Number of Workers | |---|---| | 200-250 | 12 | | 250-300 | 18 | | 300-350 | 25 | | 350-400 | 20 | | 400-450 | 15 | | 450-500 | 10 | A: Step 1: Create cumulative frequency tables. Less than CF: | Daily Wages (Upper Limit) | Number of Workers | Cumulative Frequency | |---|---|---| | < 250 | 12 | 12 | | < 300 | 18 | 30 | | < 350 | 25 | 55 | | < 400 | 20 | 75 | | < 450 | 15 | 90 | | < 500 | 10 | 100 | More than CF: | Daily Wages (Lower Limit) | Number of Workers | 'More Than' Cumulative Frequency | |---|---|---| | >= 200 | 100 | 100 | | >= 250 | 88 | 88 | | >= 300 | 70 | 70 | | >= 350 | 45 | 45 | | >= 400 | 25 | 25 | | >= 450 | 10 | 10 | Step 2: Plot 'less than' Ogive using points (200,0), (250,12), (300,30), (350,55), (400,75), (450,90), (500,100). Step 3: Plot 'more than' Ogive using points (200,100), (250,88), (300,70), (350,45), (400,25), (450,10), (500,0). Step 4: Draw smooth curves for both. The intersection point of the two ogives will give the median. Read the x-coordinate of the intersection point. Final answer: The intersection point will be approximately at (330, 50). So, the median daily wage is approximately ₹330.

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