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Data Analysis


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Data Analysis

Scientists use graphs to make pictures of the data, which is a much easier way to “read” the results to answer our questions. (Note: data is a plural word, like scissors. The singular is datum.)

How to build a histogram in Excel (the example is bogus data):


How to build a Histogram:

Students need to transfer their data from their data collection sheets to a worksheet in Excel.


Length

Width

Number

Species

Common name

88.40

52.90

4026

Dasypus novemcinctus

nine-banded armadillo

91.20

57.65

4032

Dasypus novemcinctus

nine-banded armadillo

93.90

54.15

4030

Dasypus novemcinctus

nine-banded armadillo

94.70

50.95

4036

Dasypus novemcinctus

nine-banded armadillo

95.30

56.05

4040

Dasypus novemcinctus

nine-banded armadillo

97.00

59.70

4033

Dasypus novemcinctus

nine-banded armadillo

97.20

53.60

4035

Dasypus novemcinctus

nine-banded armadillo

97.90

56.30

4027

Dasypus novemcinctus

nine-banded armadillo

98.00

47.40

4038

Dasypus novemcinctus

nine-banded armadillo

98.10

64.65

4037

Dasypus novemcinctus

nine-banded armadillo

98.40

55.55

4034

Dasypus novemcinctus

nine-banded armadillo

99.20

66.55

4039

Dasypus novemcinctus

nine-banded armadillo

100.00

49.85

4028

Dasypus novemcinctus

nine-banded armadillo

100.20

52.60

4029

Dasypus novemcinctus

nine-banded armadillo

104.10

68.05

4031

Dasypus novemcinctus

nine-banded armadillo

96.91

56.40

average




 



Bins

# specimens

88 to 91

1

91 to 94

2

94 to 97

3

97 to 100

7

100 to 103

2

103 to 106

1
A histogram can be constructed by segmenting the range of the data into equal sized bins (also called segments, groups or classes). For example, if your data ranges from 88.4 to 104.1, you could have 6 equal bins of 3 consisting of (88.01 to 91) (91.01 to 94) (94.01 to 97) (97.01 to 100) (100.01 to 103) (103.01 to 106). Note that the graph doesn’t include the .01 but it is implied. Set up your data on your Excel spreadsheet as the example on the left.

Count the number of data points that reside within each bin (i.e. the specimens that fall within the range) and construct the histogram. The user (you) defines the size of the bins, by some common rule. Students can play around with the number of bins to find the optimum to display their data. That means determine the number of bins (how are you going to evenly divide the smallest to largest skull). Then count how many specimens fit into each bin.


Excel will allocate bin divisions, but they are very artificial, and the data never look right. It might take a couple of trial and errors to find the right bin size for your data, but it really is better when you do it.
To build the histogram in Excel, select “Graphs” in the “Insert” menu or the icon for building graphs. Select “Bar Graphs.” When the menu prompts you for the data range, select both of the columns pictured above. On the next menu prompt, add your graph title and the axes labels. All the other menu prompts are for the bells and whistles, and you can make the graph look the way you want.
The vertical axis of the histogram is labeled Frequency (Number of Specimens), and the horizontal axis of the histogram is labeled with the range of your response variable (Length of Skulls in mm).

What does the histogram provide?

  • The most common system response. (97 to 100mm)

  • The distribution (center, variation and shape) of the data?

  • If the data look symmetric or are they skewed to the left or right? (skewed slightly to the left, but that could be just the low numbers of specimens in our sample)

  • Do data contain outliers? If you had a skull that was 84.3, that would be very obvious. You could then verify if the skull belonged to a juvenile, or if it is actually just an outlier.

How to build a scatter plot in Excel (the example is bogus data):

Select “Graphs” in the “Insert” menu or the icon for building graphs. Select “XY Scatter.” You want the first scatter plot, the one without the lines. On step 2 of 4, the menu will prompt you for the data range. Pick the tab “Data Range.” Click in the Data Range space, and then you can select the columns you want to include in the graph. If you have an average column (as there is in this example) do not select that. Only select the measurements.


On the next menu (page 3 of 4) you can remove the unnecessary “Series” and the legend box. Add the graph title and the axes labels.

How to build a scatter plot by hand (The example is bogus data):

http://www.terragon.com/tkobrien/algebra/topics/scatterplots/scatterplots.html


A history teacher asked her students how many hours of sleep they had the night before a test. The data below shows the number of hours the student slept and their score on the exam. The graph is a scatter plot from the given data.

Student Number
1
2
3
4
5
6
7
8
9
10

Hours Slept
8
7
7
8
6
5
7
4
9
7

Test Score
83
86
74
88
76
63
90
60
89
81

     







Make a Best Fit Line for the data:
To approximate a best fit line for the data in the graph above. First sketch a line that closely fits the data. Second, locate two points on the line. They don't have to be one of the original data points. We, will choose, (4,60) and (7,80) for this example.
http://www.purplemath.com/modules/slope.htm
For older students, you can find the equation of a line using:
y = mx + b
This is called the slope-intercept because “m” is the slope and “b” gives the y-intercept. e of the most important properties of a straight line is in how it angles away from the horizontal. This concept is reflected in something called the "slope" of the line.

 
Let's take a look at the straight line y = ( 2/3 )x – 4. Its graph looks like this:

To find the slope, we will need two points from the line.



  



Pick two x's and solve for each corresponding y: If, say, x = 3, then y = ( 2/3 )(3) – 4 = 2 – 4 = –2. If, say, x = 9, then y = ( 2/3 )(9) – 4 = 6 – 4 = 2. (By the way, I picked the x-values to be multiples of three because of the fraction. It's not a rule that you have to do that, but it's a helpful technique.) So the two points (3, –2) and (9, 2) are on the line y = ( 2/3 )x – 4.

To find the slope, you use the following formula:



(Why "m" for "slope", rather than, say, "s"? The official answer is: Nobody knows.)

The subscripts merely indicate that you have a "first" point (whose coordinates are subscripted with a "1") and a "second" point (whose coordinates are subscripted with a "2"); that is, the subscripts indicate nothing more than the fact that you have two points to work with. It is entirely up to you which point you label as "first" and which you label as "second". For computing slopes with the slope formula, the important thing is that you subtract the x's and y's in the same order. For our two points, if we choose (3, –2) to be the "first" point, then we get the following:

The first y-value above, the –2, was taken from the point (3, –2) ; the second y-value, the 2, came from the point (9, 2); the x-values 3 and 9 were taken from the two points in the same order. If we had taken the coordinates from the points in the opposite order, the result would have been exactly the same value:



As you can see, the order in which you list the points really doesn't matter, as long as you subtract the x-values in the same order as you subtracted the y-values. Because of this, the slope formula can be written as it is above, or alternatively it can be written as:



Copyright ©

Let me emphasize: it does not matter which of the two formulas you use or which point you pick to be "first" and which you pick to be "second". The only thing that matters is that you subtract your x-values in the same order as you had subtracted your y-values.

Technically, the equivalence of the two slope formulas above can be proved by noting that:

y1 – y2 = –y2 + y1 = –(y2 – y1)
x1 – x2 = –x2 + x1 = –(x2 – x1)

Doing the subtraction in the so-called "wrong" order serves only to create two "minus" signs which cancel out. The upshot: Don't worry too much about which point is the "first" point, because it really doesn't matter. (And please don't send me an e-mail claiming that the order does somehow matter, or that one of the above two formulas is somehow "wrong". If you think I'm wrong, plug pairs of points into both formulas, and try to prove me wrong! And keep on plugging until you "see" that the mathematics is in fact correct.)

Let's find the slope of another line equation:


Graphing the line, it looks like this:

  




I'll pick a couple of values for x, and find I'll find the corresponding values for y. Picking x = –1, I get y = –2(–1) + 3 = 2 + 3 = 5. Picking x = 2, I get y = –2(2) + 3 = –4 + 3 = –1. Then the points (–1, 5) and (2, –1) are on the line y = –2x + 3. The slope of the line is then calculated as:

To find the actual slope of your data for a Best Fit Line:

m = (80-60) / (7-4)
m = 20 / 3
m = 20/3

y = mx + b


y = 20/3x + b
60 = (20/3)4 + b
60 = 80/3 + b
100/3 = b

Thus a best fit equation is:


y = 20/3x + 100/3




A scatter plot shows the relationship between 2 variables. We will look at three relationships between x & y: Positive, Negative and No relationship.




Data Conclusions

After your students have completed the data analysis, you can ask them to display their findings in a variety of methods, but the most creative is the scientific poster.


In this poster, students will need to do the following:

  • Find a creative way to present the experiment.

  • Have fun with this, especially the artsy students!

  • Be neat and orderly.

  • They need 4 sections plus a title in your poster.

    • Title – generally, what is this research? Can include something catchy. Some examples:

      • Walk on the Wild Side: How Mammals Move

      • Where Art Thou Ernanodon? Comparing skeletons of ground and tree sloths, anteaters, armadillos, and glyptodonts to Ernanodon to determine if it is related

      • Sloths et al. Teaching Scientific Inquiry to 3rd – 8th graders

    • Introduction – tell me about dry ice and water ice. You need to use experts’ ideas, but stated in your own words. Let me know what expert you learned about these two chemicals. I cite sources in lines like this (Shaw 2008) or (Shaw et al. 2008) for more than one author. Include a list of sources that you cite. This introduction should help me understand your particular experiment. Your students will need to spend time researching these two chemicals.

      • Dry ice: what kind of chemical is it? Is it found naturally in the environment, or is it manufactured? How does it impact our environment?

      • Water ice: what kind of chemical is it? Is it found naturally in the environment, or is it manufactured? How does it impact our environment?

    • Methods – what is the experimental design. In this section, you can list the materials and supplies you used, diagrams, drawings or photos of the design in action. Be sure to include everything so that someone who has never done this experiment could replicate your experiment.

      • Your hypothesis

      • All the materials (be specific, i.e.

        • Dry ice in approximately 2” cubes

      • How are these materials put together?

      • What do you do to make the compare these two chemicals?

      • How did you measure them?

    • Results

      • Put your data into a table (you can make it similar to your collection sheet)

      • Develop at least 1 graph of your data (suggestion, bar graph or scatter plots will both work)

        • You may draw your graph by hand but you MUST use a ruler and graph paper (free on the internet)

    • Discussion/Conclusions

      • What happened in your experiment?

      • Was your hypothesis supported or rejected?

Example of poster on a crater impact experiment:


You could opt to have your students present their research in a PowerPoint presentation. If you are interested in finding out more about this, please email me, and I will give you additional details.

As a final product, your students could write a paper (using the same sections as needed in the scientific poster).








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