Question

Given the data set below, use it to answer the following questions:

10, 20, 30, 40, 50

a. Find the standard deviation

b. Add 5 to each data set value and find the standard deviation.

c. Subtract 5 from each value and find the standard deviation.

d. Multiply by 5 on each value and find the standard deviation.

e. Divide by 5 on each value and find the standard deviation.

f. Generalize the results in parts a-e.

Answer 1

X	(X - X̄)²
10	400.000
20	100.000
30	0.000
40	100.000
50	400.000

.

	X	(X - X̄)²
total sum	150	1000.00
n	5	5

.

mean =    ΣX/n =    150.000   /   5   =   30.0000

a) sample std dev =   √ [ Σ(X - X̄)²/(n-1)] =   √   (1000/4)   =       15.8114
b)    sample std dev =   √ [ Σ(X - X̄)²/(n-1)] =   √   (1000/4)   =       15.8114

c)    sample std dev =   √ [ Σ(X - X̄)²/(n-1)] =   √   (1000/4)   =       15.8114

d)

X	(X - X̄)²
50	10000.000
100	2500.000
150	0.000
200	2500.000
250	10000.000

	X	(X - X̄)²
total sum	750	25000.00
n	5	5

mean =    ΣX/n =    750.000   /   5   =   150.0000
                      
  
sample std dev =   √ [ Σ(X - X̄)²/(n-1)] =   √   (25000/4)   =       79.0569

e)

X	(X - X̄)²
2	16.000
4	4.000
6	0.000
8	4.000
10	16.000

	X	(X - X̄)²
total sum	30	40.00
n	5	5

mean =    ΣX/n =    30.000   /   5   =   6.0000
                      
  
sample std dev =   √ [ Σ(X - X̄)²/(n-1)] =   √   (40/4)   =       3.1623

f)

on adding or subtracting some value on each value, std dev does not change.

but Multiply by any number  on each value , then new std dev = that number*std dev

and Divide by a number  on each value , then new std dev = std dev / that number