# Knoop Microhardness Experiment

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```					EMCH 361 Measurements and Instrumentation                                                                     Lab 1

Lab 1 -- Statistics/Probability (Take Home)
SUMMARY
This is a paper clip bending experiment (sample size 20). You will start the experiment in class and will continue
it after class. The data to be recorded is the number of bends necessary to break the clip. This will constitute the
statistical data. Based on the statistical data, you will calculate some statistical/probability results.
OBJECTIVES:
This homework explores statistics/probability principles using as an example the paper clip bending test.
TEXTBOOK REFERENCE:
Chapter 3, sections 1-5, 10-11, 14
EQUIPMENT:
Paper clips

1.   THEORETICAL BACKGROUND:

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EMCH 361 Measurements and Instrumentation                                                                                                                       Lab 1

1.      THEORETICAL BACKGROUND: .............................................................................................................. 1

2.      EXPERIMENTAL PROCEDURE: ............................................................................................................... 3
2.1       PRACTICE TESTS ........................................................................................................................................ 3
2.2       ACTUAL TESTS ........................................................................................................................................... 3
2.3       PROCEDURE DETAILS ................................................................................................................................. 4
3.      DATA REDUCTION ....................................................................................................................................... 5

4.      DATA INTERPRETATION ........................................................................................................................... 6
4.1    STATISTICAL FREQUENCY .......................................................................................................................... 6
4.2    STATISTICAL CHARACTERISTICS OF YOUR SAMPLE .................................................................................. 6
4.3    HISTOGRAM CHART ................................................................................................................................... 7
4.4    RELATIVE FREQUENCY AND NORMAL DISTRIBUTION CHARTS................................................................. 7
4.5    CONFIDENCE INTERVALS ........................................................................................................................... 8
4.5.1    Confidence interval for the mean ....................................................................................................... 8
4.5.2    Confidence interval for the standard deviation ................................................................................. 8
5.      TIPS AND SUGGESTIONS ........................................................................................................................... 9
5.1       CALCULATION OF STATISTICAL QUANTITIES ............................................................................................ 9
5.2       IMPORTING CHARTS INTO MS WORD ......................................................................................................... 9
6.      DISCUSSION ................................................................................................................................................. 10
6.1       CONVERGENCE OF THEORY AND EXPERIMENTS ...................................................................................... 10
6.2       SOURCES OF ERRORS ............................................................................................................................... 10
7.      CONCLUSIONS ............................................................................................................................................ 10
7.1    CONCLUSIONS ABOUT THE EXPERIMENT ................................................................................................. 10
7.2    CONCLUSIONS ABOUT THE SAMPLED SPECIMENS ................................................................................... 10
7.3    SUGGESTIONS FOR IMPROVEMENT........................................................................................................... 10
7.3.1   Suggestions for improvement of the experiment .............................................................................. 10
7.3.2   Suggestions for improvement in the data reduction ......................................................................... 10
7.3.3   Suggestions for improvement in data interpretation ........................................................................ 10

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EMCH 361 Measurements and Instrumentation                                                                 Lab 1

2.         EXPERIMENTAL PROCEDURE:

2.1         Practice Tests
Take a paper clip and bend it backwards and forward until it breaks. During the bending, count the number of
bend reversals. Write down your result. Try another one. Write down your result. If you feel confident that you
got it under control, proceed to the next step.

2.2         Actual Tests
Select 20 paper clips that will constitute your sample. The paper clips are your test specimens. Bend them until
they break. After each specimen breaks, record in Table 1 the number of bends taken to break and any comments
you may have. Then proceed with the next specimen.
Note: Do your entries in pencil. Use an eraser for corrections. Preserve a neat appearance.

Table 1                   Experimental data
(number of bends)
Statistical data
Sample #

1             8
2             9
3             9
4             9
5            10                I think it slipped from my hands
6             9
7             7
8             9
9            10                I think it slipped from my hands
10             8
11             8
12             7
13             8
14             8
15             9
16             9
17             6                I must have put my nail into it
18             7
19            10                I think it slipped from my hands
20             9

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EMCH 361 Measurements and Instrumentation                                                             Lab 1

2.3   Procedure Details
Give here details of your experimental procedure. Give sufficient information for other people to be able to

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EMCH 361 Measurements and Instrumentation                                                                   Lab 1

3.             DATA REDUCTION
Use the data recorded in Table 1 to create a table of ranked data (Table 2). Count and enter the cumulative number
of occurrences at each repeated level. (For example, if 4 samples broke after the same number of bends, say, 14
bends, count them in increasing sample order as 1, 2, 3, 4.)
Table 2                        Ranked data
Cumulative number
(number of bends)

of occurrences
Statistical data

Sample #

6               17               1
7               7                1
7               12               2
7               18               3
8               1                1
8               10               2
8               11               3
8               13               4
8               14               5
9               2                1
9               3                2
9               4                3
9               6                4
9               8                5
9               15               6
9               16               7               This is where most of the data seems to be
9               20               8
10                5                1
10                9                2
10                19               3

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EMCH 361 Measurements and Instrumentation                                                                                                                        Lab 1

4.              DATA INTERPRETATION
Use the results from Table 2 to enter data in Table 3 and calculate the statistical frequency and the cumulative
frequency. Because you have integer values, the statistical classes have integer labels. For example, the class
containing all the samples that broke after, say, 12 bends, will be labeled ‘12’. (If you were using non-integer
values, you would have to label the bin 11.5<i<=12.5)

4.1                    Statistical Frequency
Table 3                    Statistical frequency
(number of occurrences)

Cumulative frequency
Statistical frequency

Cumulative relative
(Number of bends)

Relative frequency
Statistical class

frequency

4                        0                         0                     0.0000               0.0000
5                        0                         0                     00.000               0.0000
6                        1                         1                     0.0500               0.0500
7                        3                         4                     0.1500               0.2000
8                        5                         9                     0.2500               0.4500
9                        8                        17                     0.4000               0.8500
10                        3                        20                     0.1500               1.0000
11                        0                        20                     0.0000               1.0000
12                        0                        20                     0.0000               1.0000
13                        0                        20                     0.0000               1.0000

4.2                    Statistical Characteristics of your Sample
Calculate the mean, median, max, min, and standard deviation for your data set.
Mean                                                                    8.45
median                                                                   8.5
Min                                                                       6
Max                                                                      10
Standard Deviation                                                 1.099

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EMCH 361 Measurements and Instrumentation                                                                                                               Lab 1

4.3    Histogram Chart
Draw a histogram of statistical frequency vs. number of bends to break.
9

8

7

Statistical frequency .
6

5

4

3

2

1

0
4    5           6       7        8       9       10    11   12     13
Statistical class (number of bends to break)

Figure 1   Statistical frequency (number of occurrences) vs. statistical class (number of bends to break)

4.4    Relative Frequency and Normal Distribution Charts
Draw a histogram of relative statistical frequency vs. number of bends to break Superpose the curve representing
the normal distribution with same mean and standard deviation.
Sample relative frequency                Normal distribution, Excel function NORMDIST

0.4500

0.4000
Relative frequency .

0.3500

0.3000

0.2500

0.2000

0.1500

0.1000

0.0500

0.0000
4       5       6        7       8       9    10   11   12     13

Statistical class (number of bends to break)

Figure 2   Sample relative frequency and normal distribution with same mean and standard deviation.

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EMCH 361 Measurements and Instrumentation                                                                      Lab 1

4.5       Confidence Intervals
Calculate the confidence interval of using the mean and the standard deviation values found in your experiment as
representative of population mean and standard deviation. Assume (though not entirely correct…) that your
sample can be considered a large sample.

4.5.1 Confidence interval for the mean
Assume that the sample mean, x , determined in your experiment, is going to be used to represent the population
mean, . Give the 95% confidence interval.
Solution: Use Equation (10). Then:
    Using the MS Excel function CONFIDENCE:                7.97<<8.93 or  = 8.45+/- 0.48
    By hand: the value of c is c = 0.95. Hence c/2 = 0.475. Table 3.2 give the corresponding z value as
zc/2 = 1.96. Hence, using Equation (10),
1.1                           1.1
8.45  1.96             8.45  1.96
20                              20
or
8.45  0.48    8.45  0.48
i.e.,
7.97    8.93

4.5.2 Confidence interval for the standard deviation
Assume that the sample standard deviation, Sx, determined in your experiment, is going to be used to represent the
population mean, . Give the 95% confidence interval.
Solution: Use Equation (11). Calculate  = 1 - 0.95 = 0.05. Then:
                                                     2                    2
Either use the MS Excel function CHIINV to get X 0.025 = 32.852 and X 0.975 = 8.907

    Or look up in the textbook Table 3.5 under  = 20-1 = 19, for the values of X 0.025 = 32.852 and X 0.975 =
2                    2

8.907.
Then, using Equation (11),

 20  11.12       2

 20  11.12
32.852                       8.907
0.70   2  2.58

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EMCH 361 Measurements and Instrumentation                                                                    Lab 1

5.    TIPS AND SUGGESTIONS

5.1     Calculation of Statistical Quantities
You can perform the calculation of statistical quantities in many way:
1. Using directly the formulas
2. Using the tabulated values given in the textbook
3. Using statistical functions on your pocket calculators
4. Using the built in statistical functions available in most mathematical software packages. For example,
MS Excel has most of what you need for the analysis required in this homework. Explore and enjoy the
MS Excel Help !!! You will find this much more fun and easier to use.

5.2     Importing charts into MS Word
In performing the calculations and drawing the charts, you are encouraged to use MS Excel software or
equivalent. However, your report should be in MS Word, or equivalent. When cutting and pasting from MS Excel
into Word, use Edit\Paste Special\Picture. Do not use direct cut and paste, since this will import the Excel
application file, increase the size of your Word file, and may produce problems. After pasting your picture, right-
select the ‘Format Object’ dialog box and position the picture ‘In line with the text’. (This can be found under
‘Layout’ in Word 2000; for Word 97, uncheck the float over text check box in the ‘Position’ tab).

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EMCH 361 Measurements and Instrumentation                                                                      Lab 1

6.    DISCUSSION
Discuss the main findings from your experiment.

6.1    Convergence of Theory and Experiments
Comment on how well does the experimental data fit the theoretical model, i.e., the normal distribution. Highlight

6.2    Sources of Errors
List the probable sources of uncertainty in this lab. Explain how each factor may affect the results and list them in
terms of significance.

7.    CONCLUSIONS

Tell if the experiment was or was not conducted with professionalism and if the data collected during the
experiment should or should not be credible.

7.2    Conclusions about the Sampled Specimens
Conclude if the sampled specimens behave or do not behave in accordance with expectations.

7.3    Suggestions for Improvement
All these suggestions listed below are highly valuable to the future experimentalists. Be generous in your
suggestions.

7.3.1 Suggestions for improvement of the experiment
Give suggestions about how to improve the experimental procedure and the collection of samples. Explain how
would the experiment be conducted better, given a second chance.

7.3.2 Suggestions for improvement in the data reduction
Give suggestions for improvement of the data reduction process. Suggest software or algorithms that might be
useful to future researchers, if you know of can think of any.

7.3.3 Suggestions for improvement in data interpretation
Suggest better and more effective ways of data interpretation, if you know, or can think of any.

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