Important Formulas, Rules & Key Points

• Bar Graph: A visual representation of data using vertical or horizontal bars of equal width with uniform spacing between them. Bar height represents frequency.
• Histogram: A graphical representation of continuous class intervals where class intervals are taken along the horizontal axis (X-axis) and frequencies along the vertical axis (Y-axis). There are no gaps between bars.
• Adjustment for Continuous Classes: If class intervals are inclusive (e.g., 118-126, 127-135), convert them into continuous exclusive intervals by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit.
• Frequency Polygon: A line graph drawn by joining the mid-points (class marks) of the top edges of the rectangles in a histogram, or directly using class marks: Class Mark = (Upper Limit + Lower Limit) / 2.

Question 1

A survey conducted by an organisation for the cause of illness and death among women worldwide between the ages 15-44 (in years) found the following figures (in %):

• Reproductive health conditions: 31.8%
• Neuropsychiatric conditions: 25.4%
• Injuries: 12.4%
• Cardiovascular conditions: 4.3%
• Respiratory conditions: 4.1%
• Other causes: 22.0%

Tasks: (i) Represent the information graphically. (ii) Which condition is the major cause of women's ill health and death worldwide? (iii) Try to find out two factors which play a major role in (ii) above.

Solution:

(i) Graphical Representation: We draw a Bar Graph. Take 'Causes' on the X-axis and 'Female Fatality Rate (%)' on the Y-axis. Choose a scale where 1 unit = 5%.

(ii) Major Cause: Reproductive health conditions (31.8%) is the major cause of ill health and death among women worldwide.

(iii) Two Factors: 1. Lack of proper medical facilities and healthcare support during pregnancy. 2. Lack of awareness regarding female health and nutrition.

Question 2

The following data on the number of girls (to the nearest ten) per thousand boys in different sections of Indian society is given below:

• Scheduled Caste (SC): 940
• Scheduled Tribe (ST): 970
• Non SC/ST: 920
• Backward districts: 950
• Non-backward districts: 920
• Rural: 960
• Urban: 910

Tasks: (i) Represent the information above by a bar graph. (ii) In the classroom discuss what conclusions can be arrived at from the graph.

Solution:

(i) Bar Graph Steps: Represent 'Sections of Society' on the horizontal X-axis and 'Number of girls per 1000 boys' on the vertical Y-axis. Choose a scale: 1 unit = 100 girls starting with a kink/break from 0 to 900.

(ii) Conclusions: 1. The highest number of girls per 1000 boys is in Scheduled Tribes (970). 2. The lowest number of girls per 1000 boys is in Urban areas (910). 3. Backward districts and rural areas have a higher girl ratio compared to non-backward and urban areas.

Question 3

Given below are the seats won by different political parties in the polling outcome of a state assembly election:

• Party A: 75 seats
• Party B: 55 seats
• Party C: 37 seats
• Party D: 29 seats
• Party E: 10 seats
• Party F: 37 seats

Tasks: (i) Draw a bar graph to represent the polling results. (ii) Which political party won the maximum number of seats?

Solution:

(i) Bar Graph Construction: Plot 'Political Party' on the X-axis and 'Seats Won' on the Y-axis. Scale: 1 big division = 10 seats.

(ii) Result: Political Party A won the maximum number of seats (75 seats).

Question 4

The length of 40 leaves of a plant are measured correct to one millimetre, and the obtained data is represented in the following table:

118-126: 3, 127-135: 5, 136-144: 9, 145-153: 12, 154-162: 5, 163-171: 4, 172-180: 2

Task: Draw a histogram to represent the data.

Solution:

The given class intervals are in inclusive form. Convert them into continuous continuous form by subtracting 0.5 from lower limits and adding 0.5 to upper limits:

• 117.5 - 126.5 : 3 leaves
• 126.5 - 135.5 : 5 leaves
• 135.5 - 144.5 : 9 leaves
• 144.5 - 153.5 : 12 leaves
• 153.5 - 162.5 : 5 leaves
• 162.5 - 171.5 : 4 leaves
• 171.5 - 180.5 : 2 leaves

Plot the continuous class intervals on X-axis (using a kink break from 0 to 117.5) and number of leaves on Y-axis to draw adjacent rectangles without gaps.

Aptitude Corner

Real-World Application: Histograms and frequency polygons are widely used in data science, finance, and quality control. For example, tech companies use histograms to analyze response time latencies of servers, while financial analysts use frequency polygons to study stock market volatility over given time spans.