Question

1. construct a frequency distribution and a frequency histogram. describe any patterns.

fracture times number of classes: 5
data set:
2750 2862 2885 2490 2512 2456 2554 2532 2885 2872 2601 2877
2721 2692 2888 2755 2853 2517 2867 2718 2641 2834 2466 2596
2519

Explanation:

Step1: Find the range

The range \(R\) of a data set is given by \(R=\text{Max}-\text{Min}\).
From the data set, \(\text{Max} = 2888\) and \(\text{Min}=2456\).
So, \(R=2888 - 2456=432\).

Step2: Calculate the class width

The class width \(w\) is calculated as \(w=\frac{R}{k}\), where \(k\) is the number of classes. Here \(k = 5\).
\(w=\frac{432}{5}=86.4\). We round up \(w\) to \(90\) (for simplicity in creating classes).

Step3: Define the classes

The first class: \(2450 - 2540\) (lower limit \(2450\), upper limit \(2540\)).
The second class: \(2540 - 2630\).
The third class: \(2630 - 2720\).
The fourth class: \(2720 - 2810\).
The fifth class: \(2810 - 2900\).

Step4: Count the frequencies

• For the class \(2450 - 2540\): The data values \(2456,2466,2490,2512,2517,2519,2532,2554\) give a frequency \(f_1 = 8\).
• For the class \(2540 - 2630\): The data values \(2596,2601,2641\) give a frequency \(f_2=3\).
• For the class \(2630 - 2720\): The data values \(2692,2718\) give a frequency \(f_3 = 2\).
• For the class \(2720 - 2810\): The data values \(2721,2750,2755\) give a frequency \(f_4=3\).
• For the class \(2810 - 2900\): The data values \(2834,2853,2862,2867,2872,2877,2885,2885,2888\) give a frequency \(f_5 = 9\).

Step5: Draw the histogram

On the \(x -\)axis, mark the class intervals (\(2450 - 2540\), \(2540 - 2630\), \(2630 - 2720\), \(2720 - 2810\), \(2810 - 2900\)). On the \(y -\)axis, mark the frequencies. Draw bars for each class with height equal to the frequency of the class.

Step6: Describe the pattern

The distribution is slightly skewed. The highest frequency is in the class \(2810 - 2900\), and the lowest frequencies are in the classes \(2630 - 2720\).

Answer:

The frequency distribution table is:

Class Interval	Frequency
\(2540 - 2630\)	\(3\)
\(2630 - 2720\)	\(2\)
\(2720 - 2810\)	\(3\)
\(2810 - 2900\)	\(9\)

The pattern: The data has a higher concentration in the \(2810 - 2900\) class interval compared to other intervals, and a relatively lower concentration in the \(2630 - 2720\) class interval.