MP Board · Class 10 · Mathematics · StatisticsThe following distribution gives the daily income of 50 workers of a factory. Convert the distribution to a less than type cumulative frequency distribution and draw its ogive. Also, find the median income from the graph.
Step-by-Step Solution
To convert the given frequency distribution into a 'less than type' cumulative frequency distribution, we use the upper limits of the class intervals.
| Daily Income (Class Interval) | Number of workers ($f_i$) | Less than Upper Limit | Cumulative Frequency ($cf$) |
|---|---|---|---|
| 100-120 | 12 | Less than 120 | 12 |
| 120-140 | 14 | Less than 140 | $12 + 14 = 26$ |
| 140-160 | 8 | Less than 160 | $26 + 8 = 34$ |
| 160-180 | 6 | Less than 180 | $34 + 6 = 40$ |
| 180-200 | 10 | Less than 200 | $40 + 10 = 50$ |
Steps to construct the Ogive and find the Median:
- Plotting Points: Plot the corresponding points $(120, 12)$, $(140, 26)$, $(160, 34)$, $(180, 40)$, and $(200, 50)$ on a Cartesian plane where the upper limits are taken along the x-axis and the cumulative frequencies are taken along the y-axis.
- Drawing the Curve: Join these plotted points with a smooth freehand curve to obtain the 'less than type' ogive.
- Finding the Median:
- Here, total number of workers $N = 50$.
- Locate $\frac{N}{2} = \frac{50}{2} = 25$ on the y-axis.
- Through this point (25), draw a horizontal line parallel to the x-axis to meet the ogive curve.
- From the point of intersection on the curve, draw a perpendicular down to the x-axis.
- The point where this perpendicular meets the x-axis gives the median value.
- On reading the x-coordinate corresponding to $y = 25$, we get approximately ₹138.5 as the median daily income.
💡 Study Guide: This question tests core syllabus concepts from Statistics. For formulas, key summaries, and mock exam reference guides, read the full Statistics Revision Notes.