Statistical Calculations for Pennies Lab
Graph: Class mass values for Groups of pennies (Old, 1982, New)
X = Group
Y= Mass (graph all classmates values)
Calculate the standard deviation for each group
Standard deviation, s = [
Σ
(x
i
- x
ave
)
2
/(n - 1)]
1/2
, where n = number of sample values
Put the values (x) in order
3.043
3.074
3.074
3.076
3.080
3.086
3.087
3.093
3.094
3.107
3.114
3.114
3.114
Add values SIGMA(x) = 40.156
Calculate average (arithmetic mean) = SIGMA(x)/n, where n = number of values
SIGMA(x)/n = 40.156/13 = 3.089
x
x
i
- x
ave
(x
i
- x
ave
)
2
3.043
3.043-3.089 = -0.046
0.00211
3.074
_____
_____
3.074
_____
_____
3.076
_____
_____
3.080
_____
_____
3.086
_____
_____
3.087
_____
_____
3.093
_____
_____
3.094
_____
_____
3.107
_____
_____
3.114
_____
_____
3.114
_____
_____
3.114
_____
_____
-
-
Σ
(x
i
- x
ave
)
2
-
-
_____
standard deviation, s = 0.0206
Calculator Directions
TI-82-83
Casio 9850G
To List values (x)
STAT
EDIT
Enter values
1-Var-Stats-L1
To List values (x)
Menu
STAT
Enter values in L1
To Sort
STAT2:SortA(L1)
-
To calculate Statistics
STAT
CALC
To calculate Statistics
F2 CALC
F1 VAR
Satndard deviation (x(sigma)n-1
Q test for Rejection of Outliers
R = Range = x
n
- x
1
, n = number of observations and x
n
is the largest observation and x
1
is the smallest observation
Step 1: Rank data from lowestto highest
x
1
, x
2
, x
3
, . . ., x
n-3
, x
n-2
, x
n-1
, x
n
Step 2: Calculate Q
localc
and Q
hicalc
Q
localc
= (x
2
- x
1
)/R
Q
hicalc
= (x
n
- x
n-1
)/R
Step 3: Compare Q
localc
and Q
hicalc
with values from Q
Table
Choose confidences limit desired (e.g. 90%, 95%, 99%)
If Q
calc
> Q
Table
, reject that outlier (either x
n
or x
1
)
If Q
calc
<
Q
Table
, retain that point (either x
n
or x
1
)
Step 4: Repeat Steps 1 to 3 if aditional outliers are suspected.
Critical values for Rejection Quotient Q
Reject if Q
calc
> Q
Table
Number of
Observatons
90%
Confidence
95%
Confidence
99%
Conficence
3
0.941
0.970
0.994
4
0.765
0..829
0..926
5
0.642
0.710
0.821
6
0.560
0.625
0.740
7
0.507
0.568
0.680
8
0.468
0.526
0.634
9
0.437
0.493
0.598
10
0.412
0.466
0.568
To Compare Old Pennies and New to see if there is a statistical difference between the mass values:
s
pooled
= {[(n
old
- 1)s
old
2
+ (n
new
- 1)s
new
2
/(n
old
+ n
new
- 2)][(n
old
+ n
new
)/(n
old
x n
new
]}
1/2
t
calc
= (x
ave old
- x
ave new
)/s
pooled
Compare t
calc
with t
table
. If t
calc
> t
table
, then there is a significant difference between the two sample means (averages). However, if t
calc
< t
table
, then there is no significant difference between the two sample means (averages).
The t-test table is given below:
Values of Student's t factor - probability limits
degrees
of freedom
90%
95%
98%
99%
1
6.314
12.706
31.821
63.657
2
2.920
4.303
6.965
9.925
3
2.353
3.182
4.541
5.841
4
2.132
2.776
3.747
4.604
5
2.015
2.571
3.365
4.032
6
1.943
2.447
3.143
3.707
7
1.895
2.365
2.998
3.499
8
1.860
2.306
2.896
3.355
9
1.833
2.262
2.821
3.250
10
1.812
2.228
2.764
3.169
infinite
1.64485
1.95996
2.32634
2.57582