The mean of the following distribution is 31.87. Find the value of a
Answers
Step-by-step explanation:
QUESTION
find the value of x for the following distribution whose mean is 31.87 .
\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\begin{array}{cccc}\sf x_i &\sf f_i\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}\\\sf 12&\sf \ 8 \\\\\sf 20 &\sf \ 16 \\\\\sf 27&\sf \ 48 \\\\\sf 33&\sf 90\\\\\sf x&\sf 30 \\\\\sf 54&\sf 8 \\\\\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{\bf{}}\end{array}}\end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}
x
i
12
20
27
33
x
54
f
i
8
16
48
90
30
8
ANSWER
\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\begin{array}{cccc}\sf x_i &\sf f_i&\sf f_ix_i \\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad\qquad}{}\\\sf 12&\sf \ 8 &\sf 96 \\\\\sf 20 &\sf \ 16 &\sf 320 \\\\\sf 27&\sf \ 48&\sf 1296 \\\\\sf 33&\sf 90&\sf 2970\\\\\sf x&\sf 30 &\sf 30x \\\\\sf 54&\sf 8 &\sf 432 \\\\\sf Total &\sf \Sigma{f_i}=200&\sf \Sigma{f_ix_i}=5114 + 30x\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{\bf{}}&\frac{\qquad \qquad \qquad \qquad\qquad}{}\end{array}}\end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}
x
i
12
20
27
33
x
54
Total
f
i
8
16
48
90
30
8
Σf
i
=200
f
i
x
i
96
320
1296
2970
30x
432
Σf
i
x
i
=5114+30x
Now ,
\sf mean \: = \dfrac{\sf \Sigma{f_ix_i}}{ \sf \Sigma{f_i}}mean=
Σf
i
Σf
i
x
i
\begin{gathered} \begin{gathered}\begin{gathered}\begin{gathered} \Longrightarrow\sf 31.87 = \frac{5114 + 30x}{200} \\\end{gathered}\end{gathered} \end{gathered} \end{gathered}
⟹31.87=
200
5114+30x
\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered} \Longrightarrow\sf 6374 = 5114 + 30x \\\end{gathered}\end{gathered} \end{gathered} \end{gathered}
⟹6374=5114+30x
\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered} \Longrightarrow\sf 30x=6374 - 5114 \\\end{gathered}\end{gathered} \end{gathered} \end{gathered}
⟹30x=6374−5114
\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered} \Longrightarrow\sf 30x=1260 \\\end{gathered}\end{gathered} \end{gathered} \end{gathered}
⟹30x=1260
\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered} \Longrightarrow\sf x = \dfrac{1260}{30} = 42 \\\end{gathered}\end{gathered} \end{gathered} \end{gathered}
⟹x=
30
1260
=42
\begin{gathered}\begin{gathered} \begin{gathered}\begin{gathered} \Longrightarrow \underline{ \boxed{\displaystyle \sf hence \: x = 42 }} \\ \\\end{gathered}\end{gathered}\end{gathered}\end{gathered}
⟹
hencex=42
____________________
additional information
mean of tabular data
If \sf x_1,x_2,x_3,....,x_nx
1
,x
2
,x
3
,....,x
n
are n observations with frequencies \sf f_1,f_2,f_3,....,f_nf
1
,f
2
,f
3
,....,f
n
respectively , then the mean x of these observation is given by
\sf x = \dfrac{f_1x_1 +f_2x_2 + f_3x_3 +... + f_nx_n} {f_1 +f_2 + f_3+... + f_n} = \dfrac{\Sigma{f_ix_i} }{\Sigma{f_i}}x=
f
1
+f
2
+f
3
+...+f
n
f
1
x
1
+f
2
x
2
+f
3
x
3
+...+f
n
x
n
=
Σf
i
Σf
i
x
i
where \SigmaΣ ( called sigma ) is the Greek letter representing summation.