SME
Grade 7 Math · Problem type
Compare two samples
Use this for “Which class read longer last night, and by how much?”
Diagram
Class A: 8 values, P = 20
Class B: 8 values, Q = 32
d = 32 − 20 = 12
This is an example. Type your own values to start a new problem.
n
Students in each class
example
a1
Class A, student 1
example
a2
Class A, student 2
example
a3
Class A, student 3
example
a4
Class A, student 4
example
a5
Class A, student 5
example
a6
Class A, student 6
example
a7
Class A, student 7
example
a8
Class A, student 8
example
b1
Class B, student 1
example
b2
Class B, student 2
example
b3
Class B, student 3
example
b4
Class B, student 4
example
b5
Class B, student 5
example
b6
Class B, student 6
example
b7
Class B, student 7
example
b8
Class B, student 8
example
P
Mean of class A
worked out (not typed)
Q
Mean of class B
worked out (not typed)
d
Difference of the means
worked out (not typed)
M
MAD of class A
worked out (not typed)
D
Means apart, in MADs
worked out (not typed)
Formulas
mean
of
class
A’s
n
times
=
P
(class
A’s
times
added,
then
shared
out
evenly
=
mean
of
class
A)
mean of class A’s 8 times = 20
mean
of
class
B’s
n
times
=
Q
(class
B’s
times
added,
then
shared
out
evenly
=
mean
of
class
B)
mean of class B’s 8 times = 32
the
gap
from
P
to
Q
=
d
(the
gap
from
mean
of
class
A
to
mean
of
class
B
=
difference
of
the
means)
the gap from 20 to 32 = 12
mean
distance
of
class
A’s
n
times
from
P
=
M
(mean
distance
of
class
A’s
times
from
mean
of
class
A
=
MAD
of
class
A)
mean distance of class A’s 8 times from 20 = 5
D
=
d
÷
M
(means
apart,
in
MADs
=
difference
of
the
means
÷
MAD
of
class
A)
2.4 = 12 ÷ 5
Assumptions
•
Each class is a random sample of its grade. Times are in minutes.
•
The mean is the balance point of a dot plot; the MAD is the mean distance from it.
•
A gap between the means of two or more MADs says the groups really differ.
Step-by-step
WE KNOW
n = 8 (students in each class), a1 = 10 (class A, student 1), a2 = 15 (class A, student 2), a3 = 15 (class A, student 3), a4 = 20 (class A, student 4), a5 = 20 (class A, student 5), a6 = 20 (class A, student 6), a7 = 25 (class A, student 7), a8 = 35 (class A, student 8), b1 = 20 (class B, student 1), b2 = 25 (class B, student 2), b3 = 30 (class B, student 3), b4 = 30 (class B, student 4), b5 = 30 (class B, student 5), b6 = 35 (class B, student 6), b7 = 40 (class B, student 7), b8 = 46 (class B, student 8)
FIND
mean of class A (P), mean of class B (Q), difference of the means (d), MAD of class A (M), means apart, in MADs (D)
Step 1 · Find mean of class A (P)
Use mean of class A’s n times = P (class A’s times added, then shared out evenly = mean of class A)
Add class A’s times and share them out evenly.
P
=
(10
+
15
+
15
+
20
+
20
+
20
+
25
+
35)
÷
8
10 + 15 + 15 + 20 + 20 + 20 + 25 + 35 = 160
160 ÷ 8 = 20
P
=
20
Step 2 · Find mean of class B (Q)
Use mean of class B’s n times = Q (class B’s times added, then shared out evenly = mean of class B)
Add class B’s times and share them out evenly.
Q
=
(20
+
25
+
30
+
30
+
30
+
35
+
40
+
46)
÷
8
20 + 25 + 30 + 30 + 30 + 35 + 40 + 46 = 256
256 ÷ 8 = 32
Q
=
32
Step 3 · Find difference of the means (d)
Use the gap from P to Q = d (the gap from mean of class A to mean of class B = difference of the means)
Take the smaller mean from the bigger one.
d
=
32
−
20
d
=
12
Step 4 · Find MAD of class A (M)
Use mean distance of class A’s n times from P = M (mean distance of class A’s times from mean of class A = MAD of class A)
How far each time is from the mean, on average.
M
=
(|10
−
20|
+
|15
−
20|
+
|15
−
20|
+
|20
−
20|
+
|20
−
20|
+
|20
−
20|
+
|25
−
20|
+
|35
−
20|)
÷
8
Distances from 20: 10, 5, 5, 0, 0, 0, 5, 15
10 + 5 + 5 + 0 + 0 + 0 + 5 + 15 = 40
40 ÷ 8 = 5
M
=
5
Step 5 · Find means apart, in MADs (D)
Use D = d ÷ M (means apart, in MADs = difference of the means ÷ MAD of class A)
The means are two or more MADs apart: the classes really differ.
D
=
12
5
D
=
2.4
Check
Put the numbers back in. Both sides should match.
(10 + 15 + 15 + 20 + 20 + 20 + 25 + 35) ÷ 8 = 20 ✓
(20 + 25 + 30 + 30 + 30 + 35 + 40 + 46) ÷ 8 = 32 ✓
32 − 20 = 12 ✓
(|10 − 20| + |15 − 20| + |15 − 20| + |20 − 20| + |20 − 20| + |20 − 20| + |25 − 20| + |35 − 20|) ÷ 8 = 5 ✓
2.4 = 12 ÷ 5 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.