Various problems with data collection can cause some observations to be missing.

Question

Various problems with data collection can cause some observations to be missing. Suppose a data set has 20 cases. Here are the values of the variable x for 10 of these cases:

The values for the other 10 cases are missing. One way to deal with missing data is called imputation. The basic idea is that missing values are replaced, or imputed, with values that are based on an analysis of the data that are not missing. For a data set with a single variable, the usual choice of a value for imputation is the mean of the values that are not missing.

Create a new data set with 20 cases by setting the values for the 10 missing cases to 15. Compute the mean and standard deviation for this data set.
(Fill in the blanks below)

x¯= ____ 13.3 (wrong answer) (Give your answer as a whole number.)

s= _____ 3.64 (wrong answer) (Round your answers to 4 decimal places.)

Explanation / Answer

Getting the mean, X,      
      
X = Sum(x) / n      
Summing the items, Sum(x) =    300  
As n =    20  
Thus,      
X =    15   [ANSWER, MEAN]

**************************
      
Setting up tables,      
x   x - X   (x - X)^2
17   2   4
6   -9   81
12   -3   9
14   -1   1
20   5   25
23   8   64
9   -6   36
12   -3   9
16   1   1
21   6   36
15   0   0
15   0   0
15   0   0
15   0   0
15   0   0
15   0   0
15   0   0
15   0   0
15   0   0
15   0   0
      
      
Thus, Sum(x - X)^2 =    266  
      
Thus, as       
      
s^2 = Sum(x - X)^2 / (n - 1)      
      
As n =    20  
      
s^2 =    14  
      
Thus,      
      
s =    3.7417   [ANSWER, STANDARD DEVIATION]

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