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

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Answer in Statistics and Probability for Zagar #269727

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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

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}u22c5c” , 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}u22c52=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}u22c52=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}u22c5c” , 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}u22c52=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}u22c52=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}u22c52=17”

for Percentiles:

“P_i” class = “(in/100)”th value of the observation in *cf* column

“P_i=L+\frac{(in/100)-cf}{f}u22c5c” , 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}u22c52=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}u22c52=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}u22c52=10.35”

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