To calculate the mean height based on the given frequency polygon, we follow these steps:

1. Extract the data from the frequency polygon:

The graph shows class intervals (ranges of height in cm) and their corresponding frequencies. For each interval, the midpoint is used to represent the heights when estimating the mean. From the graph:

• Class Intervals and Frequencies:

  • $100 - 125$: Frequency = 10
  • $125 - 150$: Frequency = 18
  • $150 - 175$: Frequency = 20
  • $175 - 200$: Frequency = 12
  • $200 - 225$: Frequency = 8

• Midpoints (average of lower and upper bounds of each interval):

  • $(100 + 125) \div 2 = 112.5$
  • $(125 + 150) \div 2 = 137.5$
  • $(150 + 175) \div 2 = 162.5$
  • $(175 + 200) \div 2 = 187.5$
  • $(200 + 225) \div 2 = 212.5$

2. Compute the total of midpoints weighted by their frequencies:

$\text{Weighted Sum} = (\text{midpoint} \times \text{frequency})$ $\text{Weighted Sum} = (112.5 \times 10) + (137.5 \times 18) + (162.5 \times 20) + (187.5 \times 12) + (212.5 \times 8)$ $\text{Weighted Sum} = 1125 + 2475 + 3250 + 2250 + 1700 = 10800$

3. Compute the total frequency:

$\text{Total Frequency} = 10 + 18 + 20 + 12 + 8 = 68$

4. Estimate the mean height:

$\text{Mean} = \frac{\text{Weighted Sum}}{\text{Total Frequency}}$ $\text{Mean} = \frac{10800}{68} \approx 158.8 \, \text{cm}$

Final Answer:

The estimated mean height is 158.8 cm (to 1 decimal place).

Would you like further details or have any questions?

Related Questions:

1. What is the importance of using midpoints in calculating the mean of grouped data?
2. Can this approach be applied to any frequency distribution? Why or why not?
3. How would the mean be affected if there were an additional interval with a high frequency?
4. What is the significance of calculating the mean for data analysis in sports?
5. Could we calculate other statistical measures like variance or standard deviation using this data?

Tip:

When estimating the mean for grouped data, always use the midpoints of the intervals, as it simplifies calculations while providing a reliable approximation.