(a) Find the average and SD of the list 41, 48, 50, 50, 54, 57.
(b) Which numbers on the list are within 0.5 SDs of average? within 1.5 SDs of average? 2.
(a) Both of the following lists have the same average of 50. Which one has the smaller SD, and why? No computations are necessary. (i) 50, 40, 60, 30, 70, 25, 75 (ii) 50, 40, 60, 30, 70, 25, 75, 50, 50, 50
(b) Repeat, for the following two lists. (i) 50, 40, 60, 30, 70, 25, 75 (ii) 50, 40, 60, 30, 70, 25, 75, 99, 1

Answer:

1a) Mean = 40

Standard Deviation = 5

b)i Which numbers on the list are within 0.5 SDs of average?

48, 50, 50

ii) within 1.5 SDs of average?

48, 50, 50, 54, 57

2a)Both of the following lists have the same average of 50. Which one has the smaller SD, and why? No computations are necessary.

The list with the smaller S.D = (ii) 50, 40, 60, 30, 70, 25, 75, 50, 50, 50

This because the results obtained from the second list (ii) is close to the mean , therefore, the Standard deviation is small.

b) Repeat, for the following two lists.

The list with the smaller S.D = (i) 50, 40, 60, 30, 70, 25, 75

This because the results obtained from the second list (i) is close to the mean , therefore, the Standard deviation is small.

Step-by-step explanation:

1a) Find the average and SD of the list 41, 48, 50, 50, 54, 57.

Average (Mean) = Sum of terms/Number of terms

= 41+ 48 + 50 + 50+ 54 + 57/6

= 300/6

= 50

Standard Deviation for the population= √(x - Mean)²/n

=√ (41 - 50)²+ (48- 50)² + (50- 50)² +(50- 50)² + (54 - 50)² +(57- 50)²/6

= √81 + 4 + 0 + 0 + 16 + 49/6

= √150/6

= √25

= 5

(b) Which numbers on the list are within 0.5 SDs of average?

Mean ± Standard deviation × 0.5

50 ± 5 × 0.5

50 - 2.5

= 47.5

50 + 2.5

= 52.5

The numbers on the list are within 0.5 SDs of average are number within 47.5 and 52.5 : 48, 50, 50

ii) Within 1.5 SDs of average?

Mean ± Standard deviation × 1.5

50 ± 5 × 1.5

50 - 7.5

= 42.5

50 + 7.5

= 57.5

The numbers on the list are within 1.5 SDs of average are numbers within 42.5 and 57.5: 48, 50, 50, 54, 57

2 (a) Both of the following lists have the same average of 50. Which one has the smaller SD, and why? No computations are necessary.

(i) 50, 40, 60, 30, 70, 25, 75

Mean is already give as 50

Standard Deviation = √(x - mean)²/n

= √(50 - 50) + (40- 50)² (60 -50)² +(30- 50)² + (70 - 50)² + (25 - 50)² (75 - 50)² /n

= √0 + 100 + 100 + 400+ 400 + 625 + 625/7

= √2250/7

= √321.4285714

= 17.92842914

(ii) 50, 40, 60, 30, 70, 25, 75, 50, 50, 50

Standard deviation = √(x - mean)²/n

√(50 -50)²+(40-50)² + (60 -50)²+ (30-50)² + (70 - 50)²+ (25-50)² +(75-50)² + (50 -50)² + (50 - 50)²+(50 - 50)²/10

= √0 + 100 + 100 + 400 + 400 + 625 + 625 + 0 + 0 + 0/10

√2250/10

√225

= √15

The list with the smaller S.D = Second list (ii) 50, 40, 60, 30, 70, 25, 75, 50, 50, 50

This because the results obtained from the second list (ii) is close to the mean , therefore, the Standard deviation is small.

(b) Repeat, for the following two lists.

(i) 50, 40, 60, 30, 70, 25, 75

Mean is already give as 50

Standard Deviation = √(x - mean)²/n

= √(50 - 50) + (40- 50)² (60 -50)² +(30- 50)² + (70 - 50)² + (25 - 50)² (75 - 50)² /n

= √0 + 100 + 100 + 400+ 400 + 625 + 625/7

= √2250/7

= √321.4285714

= 17.92842914

(ii) 50, 40, 60, 30, 70, 25, 75, 99, 1

Standard Deviation = √(x - mean)²/n

= √(50-50)² +( 40 -50)² +( 60 - 50)² + (30 - 50)²+( 70 - 50)² + (25 - 50)² + (75 - 50)² +(99 - 50)² +(1 -50)²/9

= √0 + 100 + 100 + 400 + 400+ 625 + 625 + 2401 +2401/9

= √7052/9

= √783.5555556

= 27.99206237

The list with the smaller S.D = (i) 50, 40, 60, 30, 70, 25, 75

This because the results obtained from the first list(i) is close to the mean , therefore, the Standard deviation is small.

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