Answer to Question #272891 in Statistics and Probability for james

Question

The table below shows the marks scored by

229

students in a statistics class test.

Marks

10  20 20  30 30  40 40  50 50  60 60  70 70 80

Frequency

12 30 x 65 y 25 17

 Given that the median mark is

46

, determine the:

(i) Values of

x

and

y

; [5 Marks]

(ii) Mode; [2 Marks]

(iii) Mean absolute deviation; [4 Marks]

(iv) Quartile deviation. [4 Marks]

Answer 1

i)

for grouped data:

median = $L+\frac{w(n/2-B)}{G}$

where:

• L is the lower class boundary of the group containing the median
• n is the total number of values
• B is the cumulative frequency of the groups before the median group
• G is the frequency of the median group
• w is the group width

median group: 40-50

cumulative frequency of median class: $B=12+30+x$

then:

median = $46=40+\frac{10(229/2-(12+30+x))}{62}$

$x=229/2-6\cdot62/10-12-30=35$

$y=n-12-30-35-65-25-17=229-12-30-35-65-25-17=45$

ii)

Mode = $L + \frac{f_m − f_{m-1}}{(f_m − f_{m-1}) + (f_m − f_{m+1})} w$

where:

• L is the lower class boundary of the modal group
• f_m-1 is the frequency of the group before the modal group
• f_m is the frequency of the modal group
• f_m+1 is the frequency of the group after the modal group
• w is the group width

modal group: 40-50

Mode = $40 + \frac{10(65 − 35)}{65-35 +65-45} =46$

iii)

Mean absolute deviation:

$dm=\frac{\sum |x_i-\overline{x}|f_i}{n}$

where f_i is frequency,

x_i is midpoint of class,

mean:

$\overline{x}=\frac{\sum f_ix_i}{n}=\frac{15\cdot12+25\cdot30+35\cdot35+45\cdot65+55\cdot45+65\cdot25+75\cdot17}{229}=45.7$

$dm=\frac{30.7\cdot12+20.7\cdot30+10.7\cdot35+0.7\cdot65+9.3\cdot45+19.3\cdot25+29.3\cdot17}{229}=12.26$

iv)

 Quartile deviation:

$QD=(Q_3-Q_1)/2$

Q_i class = (in/4)th value of the observation

$Q_i=L+\frac{in/4-cf}{f}⋅w$

where cf is cumulative frequency

Q₁ class = (229/4)=(57.25)th value of the observation

class: 30-40

$Q_1=30+\frac{57.25-77}{35}⋅10=24.36$

Q₃ class = ($229\cdot3/4$ )=(171.75)th value of the observation

class: 50-60

$Q_1=50+\frac{171.75-187}{45}⋅10=46.61$

$QD=(46.61-24.36)/2=11.13$