Question

the density of people per square kilometer for 56 countries is in the table. create a frequency distribution, relative frequency distribution, and cumulative frequency distribution using 8 classes. round to four decimal places, if necessary. the beginning lower class limit is 28 and the class width is 75.

Explanation:

Step1: Determine class limits

The first lower - class limit is 28. Since the class width is 75, the class limits are:
Class 1: 28 - 102
Class 2: 103 - 177
Class 3: 178 - 252
Class 4: 253 - 327
Class 5: 328 - 402
Class 6: 403 - 477
Class 7: 478 - 552
Class 8: 553 - 627

Step2: Calculate frequency

Count the number of data values that fall into each class.
For example, for the class 28 - 102:
The data values 44, 28, 102, 66, 36, 101, 28, 52, 37, 87, 31, 49, 33, 56, 108, 120, 117 fall into this class. So the frequency $f_1 = 17$.
Repeat this process for all classes.

Step3: Calculate relative frequency

The relative frequency of a class is given by $rf_i=\frac{f_i}{n}$, where $n = 56$ (total number of data points).
For example, for the first class with $f_1 = 17$, $rf_1=\frac{17}{56}\approx0.3036$.
Calculate $rf_i$ for all classes.

Step4: Calculate cumulative frequency

The cumulative frequency of a class is the sum of the frequencies of that class and all previous classes.
For the first class, the cumulative frequency $cf_1=f_1 = 17$.
For the second class, if $f_2$ is the frequency of the second class, then $cf_2=f_1 + f_2$.
Calculate $cf_i$ for all classes.

The completed table:

Class limits	Frequency	Relative frequency	Cumulative frequency
103 - 177	7	0.1250	24
178 - 252	5	0.0893	29
253 - 327	7	0.1250	36
328 - 402	7	0.1250	43
403 - 477	4	0.0714	47
478 - 552	4	0.0714	51
553 - 627	5	0.0893	56

Answer:

Class limits	Frequency	Relative frequency	Cumulative frequency
103 - 177	7	0.1250	24
178 - 252	5	0.0893	29
253 - 327	7	0.1250	36
328 - 402	7	0.1250	43
403 - 477	4	0.0714	47
478 - 552	4	0.0714	51
553 - 627	5	0.0893	56