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Difficulty: HardData Representation and Charts

The table below shows the distribution of ages (in years) of trees in a forest reserve, presented alongside their frequency densities for a histogram representation:

Age Interval (years)Frequency Density
0≤x<100 \le x < 103.03.0
10≤x<2510 \le x < 252.02.0
25≤x<4525 \le x < 452.252.25
45≤x<8045 \le x < 802.02.0

If the dataset is instead displayed using a pie chart, what is the sector angle representing the age interval 25≤x<4525 \le x < 45?

  1. 90∘90^\circAnswer
  2. B
    70∘70^\circ
  3. C
    60∘60^\circ
  4. D
    80∘80^\circ

Answer

The sector angle representing the age interval 25≤x<4525 \le x < 45 is 90∘90^\circ.
The frequency of each interval is found by multiplying its frequency density by its class width (3.0×10=303.0 \times 10 = 30, 2.0×15=302.0 \times 15 = 30, 2.25×20=452.25 \times 20 = 45, and 2.0×35=752.0 \times 35 = 75). The total frequency is 180180. The sector angle for 25≤x<4525 \le x < 45 (frequency 4545) is 45180×360∘=90∘\frac{45}{180} \times 360^\circ = 90^\circ.

Step-by-Step Solution

1
Calculate the class width for each interval.
Widths are: 10−0=1010 - 0 = 10, 25−10=1525 - 10 = 15, 45−25=2045 - 25 = 20, and 80−45=3580 - 45 = 35.
For histograms with unequal class widths, frequency density is defined as frequency divided by class width.
2
Calculate the frequency (ff) for each interval using f=Frequency Density×Class Widthf = \text{Frequency Density} \times \text{Class Width}.
Interval 0≤x<100 \le x < 10: f1=3.0×10=30f_1 = 3.0 \times 10 = 30.
Interval 10≤x<2510 \le x < 25: f2=2.0×15=30f_2 = 2.0 \times 15 = 30.
Interval 25≤x<4525 \le x < 45: f3=2.25×20=45f_3 = 2.25 \times 20 = 45.
Interval 45≤x<8045 \le x < 80: f4=2.0×35=75f_4 = 2.0 \times 35 = 75.
The frequency of a class in a histogram corresponds to the area of its bar.
3
Compute the total frequency (NN).
N=30+30+45+75=180N = 30 + 30 + 45 + 75 = 180.
The total frequency represents the entire dataset needed for pie chart sector calculations.
4
Calculate the sector angle ( θ \,\theta\,) for the interval 25≤x<4525 \le x < 45.
θ=f3N×360∘=45180×360∘=90∘\theta = \frac{f_3}{N} \times 360^\circ = \frac{45}{180} \times 360^\circ = 90^\circ.
The sector angle in a pie chart is proportional to the relative frequency of the class out of 360∘360^\circ.

Key Concept

Conversion between Histogram Frequency Density and Pie Chart Sector Angles
Estimated Time:2m 0s
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Topics

Data Representation and Charts

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