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.

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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

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

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?

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