Question

A histogram is drawn with three columns whose widths are in the ratio 1:2:4. The frequency densities of these classes are in the ratio 16:12:3, respectively.

A. Given that the total frequency of the data is 390, find the frequency of each class.
B. The classes with the two highest frequencies are to be merged, and a new histogram drawn. Given that the height of the column representing the merged classes is to be 30 cm, find the correct height for the remaining column.
C. Explain what problems you would encounter if asked to construct a histogram in which the classes with the two lowest frequencies are to be merged.

a) A: 60, 120, 210; B: 30 cm; C: Loss of granularity
b) A: 80, 160, 150; B: 25 cm; C: Overemphasis of outliers
c) A: 90, 180, 120; B: 35 cm; C: Data compression
d) A: 100, 200, 90; B: 40 cm; C: Distortion of central tendency

Answer 1

A: The frequencies of the three classes are 90, 180, and 120. B: The correct height for the remaining column is 35 cm. C: Merging the classes with the two lowest frequencies would result in data compression (Option C).

Step-by-step explanation:

A. Frequency Calculation:

The given ratios for the widths are 1:2:4.

The given ratios for the frequency densities are 16:12:3.

Let the common multiplier be "k."

The widths are k, 2k, 4k.

The frequencies are 16k, 24k, 12k.

The total frequency is 52k.

Given that the total frequency is 390, solve for k.

52k = 390, k = 7.5.

The frequencies for the classes are 120, 180, 90.

B. Height Calculation for Merged Classes:

The two highest frequencies are 180 and 120.

The new total frequency is 300.

The height for the merged classes is given as 30 cm.

The height for the remaining column is (300/390) * 30 = 23.08 cm (approx 25 cm).

C. Explanation for Merging Lowest Frequencies:

Merging the classes with the two lowest frequencies could lead to data compression.

Combining infrequent data points may result in the loss of detailed information, reducing the granularity of the histogram.

It may obscure variability and patterns in the data, making it difficult to interpret.

Therefore, the correct answer is:

a) A: 90, 180, 120; B: 25 cm; C: Data compression.