Question

Given the following frequency table of test grades: Round to the nearest tenth

Grades: Frequency

98 3

94 2

90 3

86 2

78 5

1. Mean

2. Median

3. Mode

4. Sample standard deviation

5. Population standard deviation

72 2

67 6

62 1

Answer 1

Solution:
Given in the question
Mean of the data set can be calculated as
Mean = (Grades*Frequency)/(Frequency) = ((98*3)+(94*2)+(90*3)+(86*2)+(78*5)+(72*2)+(67*6)+(78(5))/(3+2+3+2+5+2+6+1) = (294+188+270+172+390+144+402+62)/24 = 1922/24 = 80.1

Grades	Frequency	Grades*Frequency
98	3	294
94	2	188
90	3	270
86	2	172
78	5	390
72	2	144
67	6	402
62	1	62

Median can be calculated as
Median =(n+1)/2 = (24+1)/2 = 12.5th value i.e. 12th value is 78 and 13th value is 78 so 12.5th value is (78+78)/2 = 78. So median = 78
Mode of the data set is the value which is often occuring in the data set so mode is
Mode = 67
Sample standard deviation can be calculated as
Sample standard deviation = sqrt((Frequency(Grade - mean)^2/(n-1)) = sqrt(3*(98-80.1)^2 + 2*(94-80.1)^2 + 3*(90-80.1)^2 + 2*(86-80.1)^2 + 5*(78-80.1)^2 + 2*(72-80.1)^2 + 6*(67-80.1)^2 + 1*(62-80.1)^2)/(24-1)) = sqrt((961.23+386.42+294.03+69.62+22.05+131.22+1029.66+327.61)/23) = sqrt(3221.84/23) = 11.8

Grades	Frequency	Grades*Frequency	Grade - Mean	Frequency*(Grade -mean)^2
98	3	294	17.9	961.23
94	2	188	13.9	386.42
90	3	270	9.9	294.03
86	2	172	5.9	69.62
78	5	390	-2.1	22.05
72	2	144	-8.1	131.22
67	6	402	-13.1	1029.66
62	1	62	-18.1	327.61

Population standard deviation = sqrt((Frequency(Grade - mean)^2/n) = sqrt(3*(98-80.1)^2 + 2*(94-80.1)^2 + 3*(90-80.1)^2 + 2*(86-80.1)^2 + 5*(78-80.1)^2 + 2*(72-80.1)^2 + 6*(67-80.1)^2 + 1*(62-80.1)^2)/24) = sqrt((961.23+386.42+294.03+69.62+22.05+131.22+1029.66+327.61)/24) = sqrt(3221.84/24) = 11.6