Let’s solve each problem step by step.
We’re told:
Mean = 45
Standard deviation (SD) = 8.3
To find a value that is “X standard deviations from the mean”, we use:
Value = Mean + (Number of SDs × Standard Deviation)
Note: If it’s negative standard deviations, we subtract.
---
Problem 1: -1 standard deviation from the mean
= 45 + (-1 × 8.3)
= 45 - 8.3
=
36.7
---
Problem 2: +3 standard deviations from the mean
= 45 + (3 × 8.3)
= 45 + 24.9
=
69.9
---
Problem 3: -1 standard deviation from the mean ← Same as Problem 1!
= 45 - 8.3
=
36.7
*(Maybe this is a typo in the worksheet? But we’ll answer as written.)*
---
Problem 4: -2 standard deviations from the mean
= 45 + (-2 × 8.3)
= 45 - 16.6
=
28.4
---
Now for Problems 5–10: We are to sketch normal curves and label the x-axis at ±1, ±2, ±3 standard deviations from the mean.
Since we can’t draw here, I’ll tell you exactly what numbers to write on the x-axis for each curve.
For any normal curve:
- Center (mean) goes in the middle.
- Then mark:
→ Mean - 3SD, Mean - 2SD, Mean - 1SD, Mean, Mean + 1SD, Mean + 2SD, Mean + 3SD
Let’s calculate those for each:
---
Problem 5: mean = 95; SD = 12
Left side:
95 - 3×12 = 95 - 36 =
59
95 - 2×12 = 95 - 24 =
71
95 - 1×12 = 95 - 12 =
83
Center:
95
Right side:
95 + 12 =
107
95 + 24 =
119
95 + 36 =
131
So label x-axis:
59, 71, 83, 95, 107, 119, 131
---
Problem 6: mean = 100; SD = 15
Left:
100 - 45 =
55
100 - 30 =
70
100 - 15 =
85
Center:
100
Right:
100 + 15 =
115
100 + 30 =
130
100 + 45 =
145
Label:
55, 70, 85, 100, 115, 130, 145
---
Problem 7: mean = 60; SD = 6
Left:
60 - 18 =
42
60 - 12 =
48
60 - 6 =
54
Center:
60
Right:
60 + 6 =
66
60 + 12 =
72
60 + 18 =
78
Label:
42, 48, 54, 60, 66, 72, 78
---
Problem 8: mean = 23.8; SD = 5.2
Left:
23.8 - 3×5.2 = 23.8 - 15.6 =
8.2
23.8 - 2×5.2 = 23.8 - 10.4 =
13.4
23.8 - 5.2 =
18.6
Center:
23.8
Right:
23.8 + 5.2 =
29.0
23.8 + 10.4 =
34.2
23.8 + 15.6 =
39.4
Label:
8.2, 13.4, 18.6, 23.8, 29.0, 34.2, 39.4
---
Problem 9: mean = 676; SD = 60
Left:
676 - 180 =
496
676 - 120 =
556
676 - 60 =
616
Center:
676
Right:
676 + 60 =
736
676 + 120 =
796
676 + 180 =
856
Label:
496, 556, 616, 676, 736, 796, 856
---
Problem 10: mean = 54.2; SD = 12.3
Left:
54.2 - 3×12.3 = 54.2 - 36.9 =
17.3
54.2 - 2×12.3 = 54.2 - 24.6 =
29.6
54.2 - 12.3 =
41.9
Center:
54.2
Right:
54.2 + 12.3 =
66.5
54.2 + 24.6 =
78.8
54.2 + 36.9 =
91.1
Label:
17.3, 29.6, 41.9, 54.2, 66.5, 78.8, 91.1
---
Final Answer:
1. 36.7
2. 69.9
3. 36.7
4. 28.4
5. Label x-axis: 59, 71, 83, 95, 107, 119, 131
6. Label x-axis: 55, 70, 85, 100, 115, 130, 145
7. Label x-axis: 42, 48, 54, 60, 66, 72, 78
8. Label x-axis: 8.2, 13.4, 18.6, 23.8, 29.0, 34.2, 39.4
9. Label x-axis: 496, 556, 616, 676, 736, 796, 856
10. Label x-axis: 17.3, 29.6, 41.9, 54.2, 66.5, 78.8, 91.1
Parent Tip: Review the logic above to help your child master the concept of normal distribution worksheet.
