a) Calculate the Arithmetic Mean from the following data: 0–5 (3), 5–10 (12), 10–25 (25), 25–50 (10) b) Draw a histogram and a frequency polygon from the following data: 10–20 (10), 20–30 (20), 30–40 (30), 40–50 (25), 50–60 (10)

Introduction

Statistical tools like the Arithmetic Mean and graphical tools like histograms and frequency polygons help in summarizing and visualizing data. They provide insights into the central tendency and distribution of a dataset. In this answer, we calculate the arithmetic mean from grouped data and construct the histogram and frequency polygon for a different frequency distribution.

a) Calculation of Arithmetic Mean

Given Data:

Since class intervals are unequal, we use the formula for weighted mean:

Step 1: Find midpoints (x)

• 0–5: (0+5)/2 = 2.5
• 5–10: (5+10)/2 = 7.5
• 10–25: (10+25)/2 = 17.5
• 25–50: (25+50)/2 = 37.5

Step 2: Calculate f × x

Total frequency (Σf) = 3+12+25+10 = 50
Total f × x = 7.5 + 90 + 437.5 + 375 = 910

Arithmetic Mean (x̄) = Σ(f×x) / Σf = 910 / 50 = 18.2

b) Draw a Histogram and Frequency Polygon

Given Data:

Steps to Draw Histogram:

1. On the X-axis, mark class intervals: 10–20, 20–30, …, 50–60.
2. On the Y-axis, plot the frequencies.
3. Draw bars for each class with height corresponding to its frequency.

Steps to Draw Frequency Polygon:

1. Calculate midpoints: 15, 25, 35, 45, 55
2. Plot points (15,10), (25,20), (35,30), (45,25), (55,10)
3. Join the points with lines.
4. Extend to (5,0) and (65,0) for boundary closure.

Conclusion

We calculated the arithmetic mean as 18.2, providing a central value of sales per salesperson. Histograms and frequency polygons are useful visual tools to analyze and interpret distributions, helping in comparing data classes and spotting trends or patterns.