Question

The following table gives the weekly wages in rupees in a certain commercial

organization.

Weekly wages Frequency

30 – 32 12

32 – 34 29

34 – 36 35

36 – 38 30

38 – 40 49

40 – 42 62

42 – 44 39

44 – 46 20

46 – 48 21

48 – 50 23

Find mean, median and mode of above data.​

Answer 1

Answer:

The mean of the distribution is 40.03 Rs.

The median of the distribution is 40.16 Rs.

The mode of the distribution is 40.72 Rs.

Step-by-step-explanation:

We have given the frequency distribution of weekly wages of an organisation.

We have to find the mean, median and mode of the data.

Now,

$\displaystyle{\begin{array}{|c|c|c|c|}\cline{1-4}\bf\:Class\:(\:Weekly\:wages\:in\:Rs\:) & \bf\:Class\:Mark\:(\:x_i\:) & \bf\:Frequency\:(\:f_i\:) & \bf\:x_i\:f_i\\\cline{1-4}\sf\:30\:-\:32 & \sf\:31 & \sf\:12 & \sf\:372\\\cline{1-4}\sf\:32\:-\:34 & \sf\:33 & \sf\:29 & \sf\:957\\\cline{1-4}\sf\:34\:-\:36 & \sf\:35 & \sf\:35 & \sf\:1225\\\cline{1-4}\sf\:36\:-\:38 & \sf\:37 & \sf\:30 & \sf\:1110\\\cline{1-4}\sf\:38\:-\:40 & \sf\:39 & \sf\:49 & \sf\:1911\\\cline{1-4}\sf\:40\:-\:42 & \sf\:41 & \sf\:62 & \sf\:2542\\\cline{1-4}\sf\:42\:-\:44 & \sf\:43 & \sf\:39 & \sf\:1677\\\cline{1-4}\sf\:44\:-\:46 & \sf\:45 & \sf\:20 & \sf\:900\\\cline{1-4}\sf\:46\:-\:48 & \sf\:47 & \sf\:21 & \sf\:987\\\cline{1-4}\sf\:48\:-\:50 & \sf\:49 & \sf\:23 & \sf\:1127\\\cline{1-4}& & \sf\:N\:=\:\sum\:f_i\:=\:320 & \sf\:\sum\:x_i\:f_i\:=\:12808\\\cline{1-4}\end{array}}$

Now, we know that,

$\displaystyle{\pink{\sf\:Mean\:\overline{X}\:=\:\dfrac{\sum\:x_i\:f_i}{\sum\:f_i}}\sf\:\:\:-\:-\:[\:Formula\:]}\\\\\\\implies\sf\:Mean\:\overline{X}\:=\:\cancel{\dfrac{12808}{320}}\\\\\\\implies\sf\:Mean\:\overline{X}\:=\:\cancel{\dfrac{6404}{160}}\\\\\\\implies\sf\:Mean\:\overline{X}\:=\:\cancel{\dfrac{3202}{80}}\\\\\\\implies\sf\:Mean\:\overline{X}\:=\:\cancel{\dfrac{1601}{40}}}\\\\\\\implies\sf\:Mean\:\overline{X}\:=\:\cancel{\dfrac{800.5}{20}}\\\\\\\implies\sf\:Mean\:\overline{X}\:=\:\cancel{\dfrac{400.25}{10}}\\\\\\\implies\sf\:Mean\:\overline{X}\:=\:40.025\\\\\\\implies\boxed{\red{\sf\:Mean\:\overline{X}\:\approx\:40.03\:Rs}}}$

The mean of the distribution is 40.03 Rs.

─────────────────────

Now,

$\displaystyle{\begin{array}{|c|c|c|}\cline{1-3}\bf\:Class\:(\:Weekly\:wages\:in\:Rs\:) & \bf\:Frequency\:(\:f\:) & \bf\:Cumulative\:frequency\:(\:cf\:)\\\cline{1-3}\sf\:30\:-\:32 & \sf\:12 & \sf\:12\\\cline{1-3}\sf\:32\:-\:34 & \sf\:29 & \sf\:41\\\cline{1-3}\sf\:34\:-\:36 & \sf\:35 & \sf\:76\\\cline{1-3}\sf\:36\:-\:38 & \sf\:30 & \sf\:106\\\cline{1-3}\sf\:38\:-\:40 & \sf\:49 & \sf\:155\:\rightarrow\:cf\\\cline{1-3}\boxed{\sf\:40\:-\:42}\sf\:\to\:Median\:class & \sf\:62\:\to\:f & \sf\:217\\\cline{1-3}\sf\:42\:-\:44 & \sf\:39 & \sf\:256\\\cline{1-3}\sf\:44\:-\:46 & \sf\:20 & \sf\:276\\\cline{1-3}\sf\:46\:-\:48 & \sf\:21 & \sf\:297\\\cline{1-3}\sf\:48\:-\:50 & \sf\:23 & \sf\:320\\\cline{1-3} & \sf\:N\:=\:\sum\:f\:=\:320 &\\\cline{1-3}\end{array}}$

Now,

$\displaystyle{\bullet\sf\:Lower\:class\:limit\:of\:median\:class\:(\:L\:)\:=\:40}\\\\\\\displaystyle{\bullet\sf\:Sum\:of\:frequencies\:(\:N\:)\:=\:320}\\\\\\\displaystyle{\bullet\sf\:Class\:interval\:of\:median\:class\:(\:h\:)\:=\:2}\\\\\\\displaystyle{\bullet\sf\:Frequency\:of\:median\:class\:(\:f\:)\:=\:63}\\\\\\\displaystyle{\bullet\sf\:Cumulative\:frequency\:of\:class\:preceding\:median\:class\:(\:cf\:)\:=\:155}$

Now, we know that,

$\displaystyle{\pink{\sf\:Median\:=\:L\:+\:\left(\:\dfrac{\dfrac{N}{2}\:-\:cf}{f}\:\right)\:\times\:h}\sf\:\:\:-\:-\:[\:Formula\:]}\\\\\\\displaystyle{\implies\sf\:Median\:=\:40\:+\:\left(\:\dfrac{\cancel{\dfrac{320}{2}}\:-\:155}{62}\:\right)\:\times\:2}\\\\\\\displaystyle{\implies\sf\:Median\:=\:40\:+\:\left(\:\dfrac{160\:-\:155}{62}\:\right)\:\times\:2}\\\\\\\displaystyle{\implies\sf\:Median\:=\:40\:+\:\dfrac{5}{\cancel{62}}\:\times\:\cancel{2}}\\\\\\\displaystyle{\implies\sf\:Median\:=\:40\:+\:\dfrac{5}{31}}\\\\\\\displaystyle{\implies\sf\:Median\:=\:\dfrac{40\:\times\:31\:+\:5}{31}}\\\\\\\displaystyle{\implies\sf\:Median\:=\:\dfrac{1240\:+\:5}{31}}\\\\\\\displaystyle{\implies\sf\:Median\:=\:\cancel{\dfrac{1245}{31}}}\\\\\\\displaystyle{\implies\sf\:Median\:=\:40.161}\\\\\\\displaystyle{\implies\boxed{\red{\sf\:Median\:=\:40.16\:Rs}}}$

The median of the distribution is 40.16 Rs.

─────────────────────

Now,

NOTE: Refer to the attachment for the steps.

The mode of the distribution is 40.72 Rs.

Answer 2

Answer:

The following table gives the weekly wages in rupees in a certain commercial

organization.

Weekly wages Frequency

30 – 32 12

32 – 34 29

34 – 36 35

36 – 38 30

38 – 40 49

40 – 42 62

42 – 44 39

44 – 46 20

46 – 48 21

48 – 50 23

The following table gives the weekly wages in rupees in a certain commercial

organization.

Weekly wages Frequency

30 – 32 12

32 – 34 29

34 – 36 35

36 – 38 30

38 – 40 49

40 – 42 62

42 – 44 39

44 – 46 20

46 – 48 21

48 – 50 23

Find mean, median and mode of above data.

Find mean, median and mode of above data.