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 - class lower limit is 28 and class width is 75. So the class limits are:
1st class: 28 - 102, 2nd class: 103 - 177, 3rd class: 178 - 252, 4th class: 253 - 327, 5th class: 328 - 402, 6th class: 403 - 477, 7th class: 478 - 552, 8th class: 553 - 627.

Step2: Calculate frequency

Count the number of data values that fall into each class. For example, for the class 28 - 102, we count the number of values in the data set that are greater than or equal to 28 and less than 102.

Step3: Calculate relative frequency

The relative frequency of a class is calculated as $\text{Relative Frequency}=\frac{\text{Frequency of the class}}{\text{Total number of data values}}$. Here, the total number of data values is $n = 56$.

Step4: Calculate cumulative frequency

The cumulative frequency of a class is the sum of the frequencies of that class and all previous classes. The cumulative frequency of the first - class is equal to its frequency. For subsequent classes, we add the frequency of the current class to the cumulative frequency of the previous class.

Class limits	Frequency	Relative frequency	Cumulative frequency
103 - 177	7	$\frac{7}{56}=0.1250$	$17 + 7=24$
178 - 252	5	$\frac{5}{56}\approx0.0893$	$24 + 5=29$
253 - 327	7	$\frac{7}{56}=0.1250$	$29+7 = 36$
328 - 402	8	$\frac{8}{56}\approx0.1429$	$36 + 8=44$
403 - 477	4	$\frac{4}{56}\approx0.0714$	$44+4 = 48$
478 - 552	5	$\frac{5}{56}\approx0.0893$	$48 + 5=53$
553 - 627	3	$\frac{3}{56}\approx0.0536$	$53+3 = 56$

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	8	0.1429	44
403 - 477	4	0.0714	48
478 - 552	5	0.0893	53
553 - 627	3	0.0536	56