SME
Grade 9 Math · Problem type
Standard deviation
Use this for “Find the mean and the standard deviation of 1, 3, 3, 5, 5, 7, 7, 9.” From 3 to 8 values.
Diagram
8 values in order: 1, 3, 3, 5, 5, 7, 7, 9.
The two middle values are both 5, so the median is 5.
This is an example. Type your own values to start a new problem.
n
Number of values
example
x₁
Value 1
example
x₂
Value 2
example
x₃
Value 3
example
x₄
Value 4
example
x₅
Value 5
example
x₆
Value 6
example
x₇
Value 7
example
x₈
Value 8
example
x̄
Mean
worked out (not typed)
S
Sum of squared deviations
worked out (not typed)
σ
Standard deviation
worked out (not typed)
Formulas
x̄
=
sum
of
the
n
values
n
5
=
sum
of
the
8
values
8
S
=
sum
of
the
squared
distances
of
the
n
values
from
x̄
48 = sum of the squared distances of the 8 values from 5
σ
=
√(
S
n
)
2.4495
=
√(
48
8
)
Assumptions
•
The standard deviation is the typical distance of the values from the mean.
•
σ = √(sum of squared deviations ÷ n): square each distance, add, divide by n, take the root.
•
In a mound-shaped data set, about two-thirds of the values lie within one standard deviation of the mean.
•
This is the population standard deviation σ (divide by n); a sample’s s divides by n − 1.
Step-by-step
WE KNOW
n = 8, x₁ = 1, x₂ = 3, x₃ = 3, x₄ = 5, x₅ = 5, x₆ = 7, x₇ = 7, x₈ = 9
FIND
mean (x̄), sum of squared deviations (S), standard deviation (σ)
Step 1 · Find mean (x̄)
Use x̄ = sum of the n values ÷ n
Add the values, then divide by how many there are.
x̄
=
(
x₁
+
x₂
+
x₃
+
x₄
+
x₅
+
x₆
+
x₇
+
x₈
)
÷
n
x̄
=
(1
+
3
+
3
+
5
+
5
+
7
+
7
+
9)
÷
8
1 + 3 + 3 + 5 + 5 + 7 + 7 + 9 = 40
40
8
=
5
x̄
=
5
Step 2 · Find sum of squared deviations (S)
Use S = sum of the squared distances of the n values from x̄
Each value’s distance from the mean, squared, all added.
S
=
(
x₁
−
x̄
)
2
+
(
x₂
−
x̄
)
2
+
(
x₃
−
x̄
)
2
+
(
x₄
−
x̄
)
2
+
(
x₅
−
x̄
)
2
+
(
x₆
−
x̄
)
2
+
(
x₇
−
x̄
)
2
+
(
x₈
−
x̄
)
2
S
=
(1 − 5)
2
+
(3 − 5)
2
+
(3 − 5)
2
+
(5 − 5)
2
+
(5 − 5)
2
+
(7 − 5)
2
+
(7 − 5)
2
+
(9 − 5)
2
Squared distances: 16, 4, 4, 0, 0, 4, 4, 16
16 + 4 + 4 + 0 + 0 + 4 + 4 + 16 = 48
S
=
48
Step 3 · Find standard deviation (σ)
Use σ = √(S ÷ n)
Divide
by
n
,
then
take
the
square
root.
σ
=
√(
48
8
)
σ
=
√
6
σ
=
2.4495
Check
Put the numbers back in. Both sides should match.
(1 + 3 + 3 + 5 + 5 + 7 + 7 + 9) ÷ 8 = 5 ✓
(1 − 5)
2
+
(3 − 5)
2
+
(3 − 5)
2
+
(5 − 5)
2
+
(5 − 5)
2
+
(7 − 5)
2
+
(7 − 5)
2
+
(9 − 5)
2
=
48
✓
2.4495
≈
√(
48
8
)
✓
Topic check
See what you know: 5 questions on this topic
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