Question

1. Find the mean, median, mode, population standard deviation and variance of the given data:

Items: 3,5,6,9,10,12,15

Frequency: 1,4,2,12,5,4,2

2.Find the mean, median, mode, sample standard deviation and variance of the discrete frequency distribution:

Items: 2,5,6,7,12

Frequency:1,3,10,4,2

Answer 1

The given frequency tables tell us how many times a particular item appears in our data. Ex.

In question number 1, Item 3 appears once, item 5 appears four times and so on.

In order to find the mean, we need to add up all the data and divide by the number of observations, i.e. Sum of all frequencies.

Mode is the item that appears the highest number of times, i.e. the one with the highest frequency.

Median is the item that occurs in the middle when they are sorted in ascending or descending order.

Population variance formula is

where x is the item and n is the number of items.

Population standard deviation is the square root of the population variance.  

1.

The total number of items is 30.

Mean = (sum of all items)/(total number of items)

= (Sum of item*frequency)/(total number of items)

= 271/30 = 9.0333

Median :

The items are already sorted in an ascending order. Since the total number of items is 30, the median is the average of (30/2)th item and (30/2+1)th item.

= (15th item + 16th item)/2 = (9+9)/2 = 9

MODE = Since 9 has the highest frequency (occurs most number of times), it is the mode.

VARIANCE :

Using the formula, we get

Variance =

STD. DEVIATION:

= (Variance)^0.5 = (7.6989)^0.5 = 2.7747

2.

MEAN = (sum of all items)/(total items)

= 129/20 = 6.45

MEDIAN: Since the number of items is an even number, the median will be the average of (20/2)th item and (20/2+1)th item.

= (6+6)/2 = 6

MODE : 6 appears the maximum number of times (10). So 6 is the mode.

POPULATION VARIANCE:

Using the same formula as above,

POPULATION STD. DEVIATION

= (4.5475)^0.5 = 2.1325