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There are 10 observations arranged in ascending order as given below. 45, 47, 50, 52, x, x + 2, 60, 62, 63, 74. The median of these observations is 53. Find the value of x. Also find the mean and the mode of the data.

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#### Solution

The 10 observations are arranged in ascending order as given below.

45, 47, 50, 52, x, x + 2, 60 ,62 ,63 ,74.

Since, the number of observation is 10, which is an even number and,

The median of the given observations is 53.

⇒ Median = 53

⇒ `1/2[(10/2)^"th" "observation" + (10/2+1)^"th" "observation"]=53`

⇒ `1/2[5^"th" "observation" +6^"th" "observation"]=53`

⇒ `1/2[x+x+2]=53`

⇒ 2x + 2 = 53 × 2

⇒ 2x = 106 - 2

⇒ x =`104/2`

∴ x = 52

Mean of the given data = `(45+47+50+52+52+54+60+62+63+74)/10`

= `559/10`

= 55.9

Since, the observation that is occurring the maximum is 52.

So, the mode of the data is 52.

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