Answer to Question #269727 in Statistics and Probability for Zagar

Question

Students per Teacher: The following frequency distribution shows the average number of students per teacher in the 50 states of the United States

Class Limit Frequency

9 - 11 1

12 - 14 17

15 - 17 13

18 - 20 8

21 - 23 2

24 - 26 1

Total 50

Determine:

• Arithmetic Mean

• Median

• All Quartiles

• D3, D5 and D7

• P37, P50, P8

Answer 1

x is midpoint of class, f is frequency of class

then:

mean = $\sum x_i f_i/n=\frac{10\cdot1+13\cdot17+16\cdot13+19\cdot8+22\cdot2+25\cdot1}{50}=13.2$

median = $L+\frac{n/2-B}{G}w$

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 is 15-17

median = $15+\frac{25-18}{13}\cdot2=16.08$

c - class width

f - frequency

cf - cumulative frequency

for Quartiles:

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

$Q_i=L+\frac{(in/4)-cf}{f}⋅c$ , where i=1,2,3

$Q_1$ class:

Class with $(50/4)th=(12.5)th$ value of the observation in cf column

this class is 12-14

$Q_1=12+\frac{(50/4)-18}{17}⋅2=11.35$

$Q_2$ = median = $16.08$

$Q_3$ class:

Class with $(50\cdot3/4)th=(37.5)th$ value of the observation in cf column

this class is 18-20

$Q_3=18+\frac{(150/4)-39}{8}⋅2=17.625$

for Deciles:

$D_i$ class = $(in/10)$th value of the observation in cf column

$D_i=L+\frac{(in/10)-cf}{f}⋅c$ , where i=1,2,3

$D_3$ class:

Class with $(50\cdot3/10)th=(15)th$ value of the observation in cf column

this class is 12-14

$D_3=12+\frac{(150/10)-18}{17}⋅2=11.65$

$D_5$ class:

Class with $(50\cdot5/10)th=(25)th$ value of the observation in cf column

this class is 15-17

$D_5=15+\frac{(250/10)-31}{13}⋅2=14.08$

$D_7$ class:

Class with $(50\cdot7/10)th=(35)th$ value of the observation in cf column

this class is 18-20

$D_7=18+\frac{(350/10)-39}{8}⋅2=17$

for Percentiles:

$P_i$ class = $(in/100)$th value of the observation in cf column

$P_i=L+\frac{(in/100)-cf}{f}⋅c$ , where i=1,2,3

$P_{37}$ class:

Class with $(50\cdot37/100)th=(18.5)th$ value of the observation in cf column

this class is 15-17

$P_{37}=15+\frac{(50\cdot37/10)-31}{13}⋅2=13.08$

$P_{50}$ class:

Class with $(50\cdot50/100)th=(25)th$ value of the observation in cf column

this class is 15-17

$P_{50}=15+\frac{(50\cdot50/10)-31}{13}⋅2=14.08$

$P_{8}$ class:

Class with $(50\cdot8/100)th=(4)th$ value of the observation in cf column

this class is 12-14

$P_{8}=12+\frac{(50\cdot8/10)-18}{17}⋅2=10.35$