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1 Statistical Analysis of Length Measurements

In order to understand the statistical analysis of scientific data from a measurement, two sets of

mock data, A and B, shown in Tables 1 and 3, are considered. Each data set has 30 measurements,

Li

, of the length of a sample in centimeters [cm]. For each data set, the mean (or average value) is

obtained from

L¯ =

Sum of All measurements

Number of Measurements =

PLi

N

.

To estimate the precision of these data sets, the statistical uncertainty, σ, is obtained from

σ =

sP(Li − L¯)

2

N − 1

.

From the results from Tables 1 and 3, the length of the sample is measured

LA = L¯

A ± σA = ± cm from Data Set A

and

LB = L¯

B ± σB = ± cm from Data Set B.

2 Histogram – Counting Events

To make a histogram, the maximum and minimum values of measurement are found for Data Set

A;

Lmax = cm; Lmin = cm

and, for Data Set B;

Lmax = cm; Lmin = cm

To cover all measurements from these two data sets, the bins in Tables 2 and 4, are used to count

measurements, and the frequency for each bin is obtained. Using the results in these tables, the

method below are used to make the histogram of this length measurement in Figure 1;

a) The length bins will be on the x axis and the frequencies on the y axis. Make sure to have

proper labels on the axes.

b) Add the scales with units on both axes. Each bin should be the same as those in Tables 2 and

4.

c) Using the frequency, fi of each bin in Tables 2 and 4 to make histograms in Figure 1.

d) Using the mean value of each data set, obtained in 1 and 3, draw vertical lines on both histograms.

e) Using the statistical uncertainty value of each data set, obtained in 1 and 3, draw Error Bars

on both histograms.

PHYS 2111 Experiment A. Histogram and Precision of Data 2

Data No. Length [cm] Residuals [cm] Residual Squared [cm2

]

i Li Li − L¯ (Li − L¯)

2

1 4.46

2 4.10

3 4.72

4 4.07

5 4.62

6 4.36

7 4.71

8 4.68

9 3.98

10 4.79

11 3.74

12 5.31

13 3.05

14 4.83

15 4.55

16 3.78

17 4.03

18 4.48

19 4.42

20 4.73

21 4.54

22 4.14

23 3.77

24 4.84

25 4.33

26 3.74

27 4.58

28 4.04

29 3.66

30 4.30

P

Sum Sum

Li

P(Li − L¯)

2

Average Statistical Uncertainty

L¯

A =

1

N

PLi σA =

q 1

N−1

P(Li − L¯)

2

Table 1: Measurement of the Length of a Sample: Data A

PHYS 2111 Experiment A. Histogram and Precision of Data 3

Bins, i Lmin [cm] Lmax [cm] Counting Measurements Frequency (fi)

1 2.5 2.9

2 2.9 3.3

3 3.3 3.7

4 3.7 4.1

5 4.1 4.5

6 4.5 4.9

7 4.9 5.3

8 5.3 5.7

9 5.7 6.1

10 6.1 6.5

Table 2: Counting frequencies of Data Set A for the histogram.

Figure 1: Histograms of the Length Measurements from Data Sets A and B.

PHYS 2111 Experiment A. Histogram and Precision of Data 4

Data No. Length [cm] Residuals [cm] Residual Squared [cm2

]

i Li Li − L¯ (Li − L¯)

2

1 4.69

2 4.76

3 4.02

4 3.47

5 4.52

6 3.85

7 5.18

8 5.42

9 4.07

10 3.51

11 3.69

12 5.03

13 3.10

14 3.50

15 4.08

16 4.80

17 2.60

18 4.37

19 3.44

20 5.85

21 4.26

22 4.91

23 3.86

24 5.66

25 3.57

26 4.31

27 3.82

28 5.16

29 3.18

30 3.51

P

Sum Sum

Li

P(Li − L¯)

2

Average Statistical Uncertainty

L¯

B =

1

N

PLi σB =

q 1

N−1

P(Li − L¯)

2

Table 3: Measurement of the Length of a Sample: Data B

PHYS 2111 Experiment A. Histogram and Precision of Data 5

Bins, i Lmin [cm] Lmax [cm] Counting Measurements Frequency (fi)

1 2.5 2.9

2 2.9 3.3

3 3.3 3.7

4 3.7 4.1

5 4.1 4.5

6 4.5 4.9

7 4.9 5.3

8 5.3 5.7

9 5.7 6.1

10 6.1 6.5

Table 4: Counting frequencies of Data Set B for the histogram.

3 Central Values: Median and Mode

The median is the middle value of data, when reorder data. Find the median of this data (get the

average value of the 15th and 16th greatest mesaurements.);

MedianA = [Unit : ] from Data Set A

and

MedianB = [Unit : ] from Data Set B

The mode is the most popular or frequent value of data. Obtain the mode of this data. (get the

middle value of the most frequent bin of each histogram in Figure 1;

ModeA = [Unit : ] from Data Set A

and

ModeB = [Unit : ] from Data Set B.

Using the values of median and mode obtained here, add vertical lines on the histograms in

Figure 1.

PHYS 2111 Experiment A. Histogram and Precision of Data 6

4 Discussion

• What shape do the data distributions in the histograms of Figure 1 look like?

• Discuss the precision of the measurement with the histogram. Discuss how you can estimate

the precision of the data (Statistical Uncertainty).

• Assuming that the actual length of the sample is 4.22 cm, estimate the Accuracy (or Sysmetic

Uncertainty) of both data sets.

• Discuss which data set is more accurate based on your estimate above. Explain why.

• Compare three Central Values you obtained; Mean, Median, and Mode. Are they close

reasonably to each other for both data sets?