SME
Grade 9 Math · Algebra 1
One-variable statistics: histograms, box plots, standard deviation and outliers
Use this for “Find the mean from a histogram of minutes spent on homework.”
Chart
n = 25 in 6 bins of width 10, from 0 to 60.
Mean x̄ ≈ 27.4, from the bin midpoints.
Shape: roughly symmetric.
This is an example. Type your own values to start a new problem.
f₁
Count, 0 to 10 minutes
example
f₂
Count, 10 to 20 minutes
example
f₃
Count, 20 to 30 minutes
example
f₄
Count, 30 to 40 minutes
example
f₅
Count, 40 to 50 minutes
example
f₆
Count, 50 to 60 minutes
example
n
Number of students
worked out (not typed)
x̄
Estimated mean
worked out (not typed) · min
Formulas
n
=
f₁
+
f₂
+
f₃
+
f₄
+
f₅
+
f₆
25 = 2 + 5 + 8 + 6 + 3 + 1
x̄
=
(5
×
f₁
+
15
×
f₂
+
25
×
f₃
+
35
×
f₄
+
45
×
f₅
+
55
×
f₆
)
÷
n
27.4 = (5 × 2 + 15 × 5 + 25 × 8 + 35 × 6 + 45 × 3 + 55 × 1) ÷ 25
Assumptions
•
A histogram groups values into equal bins; each bin takes its left end and leaves out its right.
•
From the counts alone the mean is an estimate: each value counts as its bin’s midpoint.
•
The estimated mean can differ from the true mean by up to half a bin width.
Step-by-step
WE KNOW
f₁ = 2, f₂ = 5, f₃ = 8, f₄ = 6, f₅ = 3, f₆ = 1
FIND
number of students (n), estimated mean (x̄)
Step 1 · Find number of students (n)
Use n = f₁ + f₂ + f₃ + f₄ + f₅ + f₆
Add the counts of all the bins.
n
=
2
+
5
+
8
+
6
+
3
+
1
n
=
25
Step 2 · Find estimated mean (x̄)
Use x̄ = (5 × f₁ + 15 × f₂ + 25 × f₃ + 35 × f₄ + 45 × f₅ + 55 × f₆) ÷ n
Count each value as its bin’s midpoint, add them all, then divide by how many there are.
x̄
=
(5
×
2
+
15
×
5
+
25
×
8
+
35
×
6
+
45
×
3
+
55
×
1)
÷
25
x̄
=
(10
+
75
+
200
+
210
+
135
+
55)
÷
25
x̄
=
685
25
x̄
=
27.4
min
Check
Put the numbers back in. Both sides should match.
25 = 2 + 5 + 8 + 6 + 3 + 1 ✓
27.4 = (5 × 2 + 15 × 5 + 25 × 8 + 35 × 6 + 45 × 3 + 55 × 1) ÷ 25 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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