Frequency Distribution and Histogram Development

The following analysis will develop a frequency distribution based on the provided dataset, create a histogram, construct a relative frequency distribution, and develop a relative frequency histogram.

Given Data

The provided data points are:

5, 3, 2, 6, 6,
7, 3, 3, 6, 7,
7, 9, 7, 5, 3,
12, 6, 10, 7, 2,
6, 8, 0, 7, 4

Step 1: Frequency Distribution

First, we will tally the occurrences of each value in the dataset.

Value	Frequency
0	1
1	0
2	3
3	6
4	4
5	7
6	7
7	8
8	3
9	1
10	2
11	0
12	2

Step 2: Histogram

To visualize the frequency distribution, we can create a histogram. Each bar will represent the frequency of each value.

Frequency Histogram:

Value
|
| 
|       ████████
|       ████████
|       ████████               ████████
|       ████████               ████████
|       ████████               ████████
|       ████████               ████████
|       ████████               ████████
|       ████████               ████████
|_______|____|_____|____|_____|____|_____|____
       0    2    4    6    8   10   12

Step 3: Relative Frequency Distribution

To create a relative frequency distribution, we calculate the relative frequency for each value by dividing its frequency by the total number of observations.

Total number of observations = 30 (counting all values)

Value	Frequency	Relative Frequency
0	1	1/30 = 0.033
2	3	3/30 = 0.100
3	6	6/30 = 0.200
4	4	4/30 = 0.133
5	7	7/30 = 0.233
6	7	7/30 = 0.233
7	8	8/30 = 0.267
8	3	3/30 = 0.100
9	1	1/30 = 0.033
10	2	2/30 = 0.067
12	2	2/30 = 0.067

Step 4: Relative Frequency Histogram

A relative frequency histogram can be visualized similarly to the frequency histogram, but with the height of each bar representing the relative frequency:

Relative Frequency Histogram:

Value
|
|
|                       ███████
|                       ███████
|                       ███████             ███████
|                       ███████             ███████
|                       ███████             ███████
|                       ███████             ███████
|                       ███████             ███████
|_______|_____|_____|_____|_____|_____|_____|
       0    .1    .2    .3    .4    .5 

Grouped Data Frequency Distribution

Assuming you have data for electronic sales for 120 months, using the 2k > n guideline, we can determine the smallest number of groups (k) to use for constructing a grouped frequency distribution.

Using the formula k=log⁡2(n)k = \log_2(n):

• Here n=120n = 120
• k≈log⁡2(120)≈log⁡10(120)log⁡10(2)≈2.0790.301≈6.9k \approx \log_2(120) \approx \frac{\log_{10}(120)}{\log_{10}(2)} \approx \frac{2.079}{0.301} \approx 6.9

Since k must be a whole number, we round up to 7. Thus, the smallest number of groups is 7.

Advantages of Relative Frequency Distribution

1. Standardization: Relative frequencies allow for comparison across different datasets or distributions regardless of sample size, as they express frequencies as proportions of the total.
2. Interpretability: Understanding relative frequencies can be more intuitive for interpreting how significant each category is in relation to the whole dataset.
3. Normalization: It helps in normalizing data and understanding patterns when working with different scales and units.
4. Data Visualization: When creating visual representations like pie charts or relative frequency histograms, it becomes easier to comprehend the distribution of data.

Overall, the construction of a relative frequency distribution offers valuable insights that can enhance data interpretation and analysis in various fields.