Penny Pinching: Statistical Treatment of Data

By: Julie Wilhelmsen

Objective:

The objective of experiment one is to see if the year a penny is produced in mint, relates to the weight of the penny. Each lab group had a set of 20 pennies, in which we collected the weight and the year of each penny. The information obtained from each lab group will help us create some statistical data. By using different calculations, I will be able to determine if there is a relation between a penny’s weight and year minted. The calculations I will use to do this are: mean, standard deviation, Q-test, and t-test. I predict that pennies that are newly produced will weigh less than pennies produced many years ago.

Data:

Table 1: Individual Lab Group Data

Penny #

Year

Mass (g)

1

1980

3.0841

2

2005

2.4948

3

1982

3.0996

4

1996

2.5038

5

1993

2.4829

6

1988

2.4724

7

1984

2.5723

8

1995

2.5191

9

1990

2.5269

10

1979

3.1068

11

1999

2.4890

12

1995

2.4901

13

1984

2.5202

14

1988

2.5154

15

1983

2.5108

16

1974

3.0823

17

1991

2.4502

18

1993

2.4807

19

1985

2.4657

20

1963

3.0910

 

 

Table 2: Class Data

Penny #

Year

Mass (g)

Penny #

Year

Mass (g)

1

1988

2.4444

41

1994

2.4624

2

1993

2.5002

42

1996

2.4743

3

1990

2.4990

43

1991

2.5226

4

1990

2.4711

44

1979

3.0753

5

1959

3.0603

45

1968

3.0454

6

1994

2.4732

46

1990

2.4991

7

1993

2.4857

47

1995

2.4516

8

2001

2.4843

48

1985

2.5118

9

1993

2.5222

49

2005

2.5108

10

2001

2.5035

50

1969

3.0703

11

1982

3.1035

51

1972

3.0694

12

1974

3.1047

52

1984

2.483

13

1989

2.5413

53

1996

2.4724

14

1989

2.4786

54

1982

3.116

15

1986

2.4923

55

1982

3.1009

16

1982

3.0563

56

1987

2.5218

17

1988

2.4533

57

1999

2.4892

18

1973

3.1142

58

1988

2.4492

19

1964

3.1085

59

1987

2.5223

20

1972

3.1025

60

1994

2.5136

21

1981

3.0091

61

1979

3.1048

22

1991

2.5374

62

1988

2.5082

23

1986

2.5418

63

1999

2.5111

24

1978

3.1921

64

1995

2.4893

25

2006

2.4812

65

1970

3.1001

26

1982

3.1121

66

2005

2.4979

27

1980

3.1286

67

1996

2.5063

28

1975

3.1243

68

1977

3.1096

29

1996

2.5083

69

1989

2.4948

30

2005

2.5084

70

1975

3.1283

31

1995

2.5006

71

1983

2.5272

32

1978

3.1169

72

1973

3.0811

33

1982

3.096

73

1995

2.4991

34

1959

3.1042

74

1978

3.1055

35

1980

3.0782

75

1989

2.5331

36

1989

2.4955

76

1982

3.1122

37

1990

2.5159

77

1986

2.4782

39

1993

2.5036

79

2003

2.5107

40

2001

2.4992

80

1994

2.517

Penny #

Year

Mass (g)

Penny #

Year

Mass (g)

81

1987

2.4966

121

1994

2.4963

82

1984

2.5722

122

1992

2.5107

83

1990

2.5288

123

1977

3.1174

84

1984

2.5001

124

2003

2.4819

85

1973

3.0682

125

1968

3.0723

86

1991

2.5163

126

2000

2.4884

87

2000

2.4757

127

1996

2.4955

88

1983

2.5606

128

1998

2.4758

89

1990

2.5261

129

1974

3.0957

90

1988

2.4743

130

2001

2.4866

91

1990

2.4639

131

1982

3.115

92

2001

2.5159

132

1979

3.0664

93

1986

2.5069

133

1988

2.9095

94

1989

2.4943

134

1992

2.4875

95

1996

2.4903

135

1975

3.1285

96

1975

3.1156

136

1984

2.5368

97

1986

2.5199

137

1963

3.0583

98

1980

3.1549

138

1994

2.5095

99

1990

2.4611

139

1998

2.5221

100

1997

2.5042

140

1980

3.0947

101

1980

3.0841

141

1994

2.4963

102

2005

2.4948

142

1992

2.5107

103

1982

3.0996

143

1977

3.1174

104

1996

2.5038

144

2003

2.4819

105

1993

2.4829

145

1968

3.0723

106

1988

2.4724

146

2000

2.4884

107

1984

2.5723

147

1996

2.4955

108

1995

2.5191

148

1998

2.4758

109

1990

2.5269

149

1974

3.0957

110

1979

3.1068

150

2001

2.4866

111

1999

2.489

151

1982

2.115

112

1995

2.4901

152

1979

3.0664

113

1984

2.5202

153

1988

2.5095

114

1988

2.5154

154

1992

2.4875

115

1983

2.5108

155

1975

2.1285

116

1974

3.0823

156

1984

2.5368

117

1991

2.4502

157

1963

3.0583

119

1985

2.4657

159

1998

2.5221

120

1963

3.091

160

1980

3.0947

 

Calculations and Graphs:

Constants Used in Calculations:

Qcrit (n=20) at 90% confidence = 0.300

This value was found on “Q” test table provided by Dr. Schug. The table was adapted from D.B. Rorabache, Anal. Chem., 63 (1981) 139.

ttable (DOF = ∞) at 99% confidence level = 2.576

This value was found on Values of Student’s t Table provided by Dr. Schug. The table was originally found in Quantitative Chemical Analysis, Seventh Edition ©2007 W.H. Freeman and Company.

 

Q-Test Calculations for Individual Data

Qcalc = Gap/Range

Gap (low) = 2.4657 – 2.4502 = 0.0155g

Gap (high) = 3.1068 – 3.0996 = 0.007g

Range (high-low) = 3.1068 – 2.4502 = 0.6566g

 

Qcalc(low) = 0.0155/0.6566 = 0.0236

Qcalc(low) < Qcrit

 

Qcalc(high) = 0.007/0.6566 = 0.11

Qcalc(high) < Qcrit

 

Q calculated in both cases (high and low) is less than Q critical (table), so we retain both values.

 

Mean of individual data=2.6479 g

Mean = ∑xi/n

∑ xi = sum of measured values

n = number of measurements

Standard deviation (s)=0.2648 g

Standard Deviation

Table 3: Low and High Frequency Table for Class Data

Range (g) low distribution # of Pennies Range (g) high distribution # of Pennies
2.100 to 2.325

2

2.800 to 2.825

0

2.325 to 2.350

0

2.825 to 2.850

0

2.350 to 2.375

0

2.850 to 2.875

0

2.375 to 2.400

0

2.875 to 2.900

0

2.400 to 2.425

0

2.900 to 2.925

1

2.400 to 2.425

0

2.900 to 2.925

1

2.425 to 2.450

2

2.925 to 2.950

0

2.450 to 2.475

13

2.950 to 2.975

0

2.475 to 2.500

40

2.975 to 3.000

0

2.500 to 2.525

37

3.000 to 3.025

1

2.525 to 2.550

10

3.025 to 3.050

1

2.550 to 2.575

3

3.050 to 3.075

11

2.575 to 2.600

0

3.075 to 3.100

13

2.600 to 2.625

0

3.100 to 3.125

21

2.625 to 2.650

0

3.125 to 3.150

3

2.650 to 2.675

0

3.150 to 3.175

1

2.675 to 2.700

0

3.175 to 3.200

1

2.700 to 2.725

0

3.200 to 3.225

0

2.725 to 2.750

0

3.225 to 3.250

0

2.7250 to 2.775

0

3.250 to 3.275

0

2.775 to 2.800

0

3.275 to 3.300

0

 

Penny Distribution

Figure 2: High Frequency of Class Data
Where the x-axis is the range in grams, and the y-axis is the number of pennies.

Penny Distribution 2

Mean and Standard Deviation Calculations

Individual Data

While performing the experiment, it became evident that there was a range of weights that can be expressed on table 1. When we calculated the Q-test, it revealed that the two highest and two lowest values were not outliers. By performing these calculations we can visibly see a difference between the highest and lowest values. After dividing these values by the range, we calculate the critical value. After calculating the critical value, it was determined that no values should be disregarded because the calculated Q-test value was less than the provided critical value.

The mean of pennies weights was 2.6479. This means that the masses were relatively close. The mean is the average weight of all the pennies in the experiment. The standard deviation is calculated to show how many of the pennies fell near the mean weight for the penny. The standard deviation was 0.2648. The standard deviation being low means that most of the pennies masses were close to the mean.

Results

After reviewing the complied data from the class, it was evident that there were a large majority of the masses were around 3 grams. If you look at the histograms ( Figure 1 and Figure 2), you can see the frequency distribution  for the masses of the pennies.  In order to tell completely analyze the data, more tests and calculation were done.

The mean and the standard deviation of the low frequency distribution of penny’s weights were calculated. The mean of the low frequency was calculated to be 2.493g and the standard deviation to be 0.0569g. The standard deviation being low means that more pennies were weighed, in which we were able to obtain more data. This means that because there is more data, the mean and the standard deviation are closer to the actual value.

The mean and the standard deviation for the high frequency distribution of penny’s weights were also calculated. The mean calculated was 3.092g and the standard deviation was 0.0385g. This puts the measurements in the range of 3.0535g to 3.1305g for the high frequency.

By calculating the two frequencies, it became evident that a T-test needed to be done to verify the data. First, the pooled standard deviation (S pool) is calculated. This was needed to plug into the T calc equation. After calculating a T-value, it was compared that value to the t-table value. Because T table was less than T calc at a 99% confidence, it can be assumed that there are two distinct distributions.

Conclusions:

It can be concluded that there is a relation between the pennies masses and the year minted. I predicted that pennies that are more newly produced would weigh less than pennies produced many years ago. This seemed to hold true while analyzing our data.

By looking at both individual and class datum, we can see two distinct average masses. In both, individual data and class data the mean was calculated, along with standard deviation, to indicate how far the measurements range is. Because the individual data’s standard deviation was higher this means that it isn’t as precise, due to less measurements. By adding more measurements, the standard deviation decreased. When I calculated the Q-test it led me to believe that no values were to be discarded in the datum. The reason some of the penny’s weights were so far off from the mean is most likely because of being measured incorrectly (human error).

I was able to conclude that the majority of that data is closer to the mean in the class data than in comparison to the individual datum. This is evident because you can slightly tell there is a Gaussian distribution in figures 1 and 2. This just shows that the measurements were replicated enough times to account for a random error, if any. When I compared the t-test datum more closely, I was able to conclude that there are two distinct distributions. The confidence interval was 99%.