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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(xi - xave)2/(n - 1)]1/2, where n = number of sample values

Put the values (x) in order

  1. 3.043
  2. 3.074
  3. 3.074
  4. 3.076
  5. 3.080
  6. 3.086
  7. 3.087
  8. 3.093
  9. 3.094
  10. 3.107
  11. 3.114
  12. 3.114
  13. 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
xi - xave
(xi - xave)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(xi - xave)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 = xn - x1, n = number of observations and xn is the largest observation and x1 is the smallest observation

Step 1: Rank data from lowestto highest

x1, x2, x3, . . ., xn-3, xn-2, xn-1, xn

Step 2: Calculate Qlocalc and Qhicalc

Qlocalc = (x2 - x1)/R
Qhicalc = (xn - xn-1)/R


Step 3: Compare Qlocalc and Qhicalc with values from QTable

Choose confidences limit desired (e.g. 90%, 95%, 99%)

If Qcalc > QTable, reject that outlier (either xn or x1)

If Qcalc < QTable, retain that point (either xn or x1)

Step 4: Repeat Steps 1 to 3 if aditional outliers are suspected.

Critical values for Rejection Quotient Q
Reject if Qcalc > QTable
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:


spooled = {[(nold - 1)sold2 + (nnew - 1)snew2/(nold + nnew - 2)][(nold + nnew)/(nold x nnew]}1/2

tcalc = (xave old - xave new)/spooled

Compare tcalc with ttable. If tcalc > ttable, then there is a significant difference between the two sample means (averages). However, if tcalc < ttable, 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