Exercises in Data Analysis

Truism: objects with the same name have the same properties (M & Ms)

Introduction

Mass production and sale of commodities promises a uniformity of product that it may or may not deliver. Everyone has heard of quality control and of objects whose performance fails to meet expectations. Everyone also knows that widely distributed commodities may appeal differently to people with different tastes and in diverse circumstances. Manufacturers know that the specific product that appeals to buyers in the Northeast may not sell well in Texas. It would be nice to test the truism that objects with the same name have the same properties by using automobiles, dolls or chickens, but the ubiquitous M&M^® candies are cheaper and easier to obtain, as well as more likely to be obtainable from great distances.

Materials Required

One bag of plain and one of peanut M&Ms for each student.

Balance for weighing samples (one for the class).

Procedure

Send home for a bag of M&M^® candies (ordinary, without nuts).

When it arrives, make a record of its source (the town and state from which it came), open the bag and sort the candies into piles by color.

Count and record the number of candies of each color.

Source of M&Ms	Blue	Brown	Green	Orange	Red	Yellow	Other	Total
Home, plain:

Get a second bag of M&M^® candies in class (source, Hanover, NH).

Open this bag and again sort, count and record the number of candies by color.

Source of M&Ms	Blue	Brown	Green	Orange	Red	Yellow	Other	Total
Hanover, plain

Get a bag of M&M^® candies with nuts (source, Hanover, NH).

Open this bag and again sort, count and record the number of candies by color.

Source of M&Ms	Blue	Brown	Green	Orange	Red	Yellow	Other	Total
Hanover, with nuts

Compile three lists of the results from others in the class (data pages follow). You will probably want to do more with the data than simply enter them in these lists in preparation for the report.

Stem and leaf diagrams or histograms.

Mean, median, and modal color distributions for the class.

Means, medians and variances for quantitative data (if collected).

Foreign (plain, without nuts)^*

Source of M&Ms	Blue	Brown	Green	Orange	Red	Yellow	Other	Total
Home:
Home:
Home:
Home:
Home:
Home:
Home:
Home:
Home:
Home:
Home:
Home:
Home:
Home:
Home:
Home:
Home:
Home:
Home:
Home:
Home:
Home:
Home:
Home:
Home:
Home:
Home:
Home:
Home:
Home:
N = _____

Local (plain, without nuts)^*

Source of M&Ms	Blue	Brown	Green	Orange	Red	Yellow	Other	Total
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
N = _____

Local (with nuts)*

Source of M&Ms	Blue	Brown	Green	Orange	Red	Yellow	Other	Total
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
Hanover
N = _____

Report

Describe the distribution of colors in candies from your three sources. How could you test to see whether the frequencies you found for the three samples differ from each other?

Compare your results with those of others in the class:

Can you find evidence of differences in frequencies among the packages of candy from Hanover (with and without nuts)?

Compare the differences in variation between class results from Hanover and foreign packages (plain candies only).

Describe the distribution of colors between the bags of plain and peanut candies.

Suggest how you can determine the relative variation in the samples at hand.

Consider whether the variations seen in one bag (in size and color) provide a good basis for describing the variation found in all the bags observed.

Calculate the observed and expected frequencies for each color candy and perform a c ²-test to see if there is an equal number of candies in each color.

Aphorism: "Like as peas in a pod"

Introduction

Many annual varieties of the family Leguminosae have been cultivated in gardens for both personal and communal use for more than 6000 years. Their edible seeds include peas, beans, lentils, peanuts, groundnuts, and vetch that grow within a pod which may itself also be edible. They dry and keep well, serving as a major source of nutrition for people everywhere. Local varieties of peas or beans have been familiar for so long that their apparent uniformity has become part of language. This exercise explores whether the aphorism, "like as peas in a pod," reflects reality or just a perception of what reality ought to be. The "null" hypothesis (H₀) states that peas in a pod do not differ. We shall test it by collecting quantitative data. From this we can ascertain whether peas in the same pod differ or whether peas in one pod differ from other peas. Note that the expression refers to peas, not grain or other legumes. Why might it be more likely to have arisen than one about the similarity of beans or maize or lentils? What does it imply about the similarity of other legumes or seeds of grains?

The first questions to answer have to do with the nature of the data, but note that the "peas" will not be the data used, but only the objects from which the data will be gathered.

What (or who) are they?

Where, when, why and how were they collected?

Perhaps the most obvious way to evaluate the similarity of peas within a pod would be to weigh and compare the weights of peas from the same pod. This procedure becomes unwieldy, however, using even a small sample size. For example, given a sample of 25 pea pods with an average number of 5 peas per pod, requires 125 individual measurements and 25 computations of means and standard deviations- a tedious process at best. Therefore, another, more efficient method should be sought for exploring the reliability of the aphorism.

One measurement of likeness is the variance, a statistic used to describe the spread of a set of values around the mean of those values. The standard deviation (square root of the variance) provides a more convenient descriptor, as it has the same units as the mean. If the aphorism holds true, the variance (or standard deviation) of the average weight of a pea within a pod would be small regardless of the number of peas in that pod. However, if the aphorism is false, that is if peas within a pod do not tend to be alike, then the variance (standard deviation) of the average weight of a pea in a pod would increase as the number of peas per pod increases. This conceptualization of variance allows measuring the contents of a pod all at once, computing the average weight per pea in that pod, and then comparing the computed variance (standard deviation) to test the aphorism.

Materials Required

Fresh peas (in pods) sufficient to provide 25 pea pods for each student.

Balance for weighing samples (one for the class).

Procedure - Part A (Preliminary)

Obtain 25 pea pods and a balance for weighing pea pods and peas.

Mark each pod randomly with a number (1-25): be sure to take the pods in random order, not arranged by size, color, or any other criterion.

Open the first pod and count the number of peas it contains.

Take peas out of the pod and weigh them as a group (not individually).

Repeat the procedure for pods numbered 2-5 only, leave the other 20 pods alone for the moment.

Pod Number	Number of Peas	Weight of Peas	Average Pea Weight
1
2
3
4
5

Preliminary Interpretation.

At this point you have selected a small sample of pea pods and sampled one quality (weight) of the contents. Note that in some pods the peas are fairly equal in size while in others the individual peas may differ dramatically in size. Is it possible that some of these peas should not be considered to be "peas" for this exercise? If you think this is the case, establish some criterion that would allow you to exclude these small ones (ovules) and count only the more developed seeds as peas. .

This process of looking at a small portion of your sample first is called "pre-testing", a method which allows the experimenter to discover any factors that need refinement before embarking on the final analysis. Pre-testing allows you to anticipate ambiguities and errors in judgement before they spoil your analysis.

Procedure - Part B (Descriptive)

Open pod #6 and count the number of peas, using the criteria you established during pre-testing.

Take the peas from pod #6 and weigh them as a group.

Repeat for remaining 20 pods

Pod Number	Number of Peas	Weight of Peas	Average Pea Weight
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25

Report

Given the data you have collected, describe the calculations and plots you think would be most appropriate for determining the likeness of peas in a pod?

Using the procedure you think best, evaluate the aphorism "like as peas in a pod."

Germination of radish seeds

Introduction

Many seeds germinate differently in the light and the dark, in soil or in a moist chamber, under different conditions of temperature or pre-germination treatment. This exercise tests the ability of a plant to germinate under defined, abnormal, and modestly different conditions. As a basis for study, germinate radish seeds in a moist chamber (petri dish with a water soaked substrate) under ambient conditions that are easy to maintain and measure (e.g., room temperature and normal illumination).

Materials Required

A room with a window open to natural light.

A desk or other object with some free space on top.

A drawer (dark when closed).

Access to a refrigerator (optional).

Two or three disposable germination chambers (petri dishes or other clear plastic containers) and absorbent paper disks/sheets (filter disks or paper towels cut to size, not toilet tissues, as they shred).

A small millimeter-ruled measuring device (sufficient for each student to have reasonable access).

A source of good water (at room temperature) that can be used to irrigate each germination chamber sufficient to keep it moist.

Procedure

Begin on a Monday (or Wednesday), planning to finish on Friday (or Sunday).

Number six positions on each filter paper using a soft pencil (#2), not a pen.

Place one piece of paper in each of the chambers and soak it with water, sufficient to thoroughly wet it, but not to leave a puddle of standing water.

Place 6 seeds in each petri dish, one at each numbered location.

Put one chamber on the desk top, one in the drawer and (optionally) one under some other condition (e.g., in a refrigerator), making a careful record of the precise conditions in which you put each chamber.

Observe each chamber once every day at the same time, taking care to keep the absorbent paper soaked with water and noting (describing and recording) the appearance of each seed or seedling in each chamber.

Pay particular care to record any seeds that fail to germinate or any changes that you see, including the color and apparent size of the cotyledons (seed leaves).

Option 1: Growth in five days

Measure and record the length of each plant at the end of five days of growth.

Try to distinguish between growth of the root (that part below the old seed coat) and the shoot (that part above it).

Calculations

Compute the mean and median size for the six seedlings in each of the two or three dishes.

Get data from the entire class and compute the group mean, median, and spreads (variance and quartile) for each condition tested.

Option 2: Rate of growth for five days

Measure and record the length of each plant on each of the five days.

Calculations

Compute the mean and median size for the six seedlings for each day in each of the three dishes.

Compute the average growth rate in each dish.

Get data from the entire class and compute the group mean, median, and spreads (variance and quartile) of growth and growth rates.

Additional options

Compare different ways of measuring the length of germinating seeds:

Compare direct measurements (as above) with measurements of a length of thread carefully laid along the seedling and then measured separately.

Follow the procedures under Option 2 (above).

Compare the consequences of desiccation:

Compare the growth of seedlings in a continually damp chamber with that of seedlings in which the paper doesn’t remain visibly moist (be careful not to let it dry out completely).

Follow the procedures under Option 2 (above).

Compare the germination of seeds using paper from different sources:

Filter paper for coffee makers (you may want to compare differences between bleached and unbleached papers).

Paper towels from different manufacturers (e.g., recycled, untreated, "wet-strong").

Other absorbent papers and newsprint (with or without ink).

Follow the procedures under Option 2 (above).

Compare the germination of seeds pretreated under different conditions:

Quickly plunged into boiling water.

Chilled in a freezer for a day prior to germination.

Soaked in vinegar prior to germination.

Follow the procedures under Option 2 (above).

Compare with the germination of seeds in soil (following the procedures under Option 2, above).

Place the seeds in a pot with fine soil (you must wait until the end of the test period to measure their length, as removing and measuring them interferes with further development).

Be careful not to break the root when removing seeds from the soil.

Report

Compare the germination, size after five days and (in Option 2) growth rates of seedlings under each condition of germination.

Graph the results.

If necessary, transform the data to make the graph linear and describe the growth of the seedlings.

What kind of transformation gives the best result, i. e., closest approximation to a straight line on the graph and to heomoscedasticity in the variation?

What can infer from the transformation?

Use Student’s t -test to see whether there is a difference in the growth of seedlings germinated under the conditions you used.

Data

Option 1: Germination of radish seeds and growth in five days, room (ambient light) conditions.

Seed Number	Day 1 (mm)	Day 2 (mm)	Day 3 (mm)	Day 4 (mm)
1
2
3
4
5
6

Option 2: Germination of radish seeds and growth in five days, dark conditions.

Seed Number	Day 1 (mm)	Day 2 (mm)	Day 3 (mm)	Day 4 (mm)
1
2
3
4
5
6

Additional Option (specify): .

Seed Number	Day 1 (mm)	Day 2 (mm)	Day 3 (mm)	Day 4 (mm)
1
2
3
4
5
6

Body Parts

Introduction

Reason suggests that many physical (and mental) properties show a high level of association: that is, they reflect some underlying property of quality that distinguishes people from each other. Following this line of thought, the dimensions of various body parts might be expected to show a close relation so that a small person would have shorter arms and legs than a taller one. But how far might these expectations extend? Would the same be true for the relative size of ear lobes, the width of shoulders, the circumference of biceps, or the length of their nose? And what of less concrete associations, such as disease resistance, spatial or odor perception, and musicality. Some of these dimensions and properties must have a greater susceptibility to environmental modification than others, even if they all show an underlying association with some index of "general size," "general quality," or "general intelligence."

Test your hypotheses (assumptions, preconceptions) by first stating them and then collecting data that measure them. Although not as controversial (interesting) as properties such as intelligence, beauty, or creativity, body parts have the advantage of easy accessibility and measurement. Use some easily measured anatomical elements to examine whether the general truism holds, that body parts reflect body size. Measure the lengths of your thumb (terminal joint), middle digit (proximal joint), forearm (ulna), and foot (heel to tip of longest toe) to give yourself data to test the general hypothesis and your particular notions of which parts vary together.

Materials Required

ruler (cm)

string

Procedure

Record your assumptions before you make any measurements about the correlation’s (positive, +; negative, -; or none, 0) that you expect to find between body length (height) and the several measured components.

Write a brief hypothesis incorporating your anticipated results. Keep the discussion simple, but include reasons for your expectations. Don’t modify this document as you continue, but keep it as a record of your preconceptions and as a basis for possible hypothesis testing.

Use the data sheet on the back of this page to record all measurements (use centimeters, estimating to the nearest tenth cm).

Record your body height.

Measure the length of the last joint of your thumb (shown in black and marked by an arrow).

Mark the distance between the middle knuckle and the thumb tip on the string and then measure that distance with the ruler.

Measure the length of your middle finger from the first knuckle to the joint between the phalanx and the 3rd metacarpal (again shown in black and marked by an arrow).

Measure the length of your ulna (outer bone in your forearm, again shown in black) using the end of the olecranon (bump on your elbow) and the outside bump on your wrist (beneath your little finger) as endpoints.

Measure the length of your foot from the tip of your largest toe to the back of your heel (note also which toe is longest).

Measure the circumference of your biceps (the muscle overlying your humerus) at its greatest width.

Obtain a list of the results from others in the class.

Report

Prepare stem and leaf diagrams or histograms of class data for each variable.

Compute an appropriate average and estimate of variation for each variable: mean ± standard deviation or median ± step.

On what basis do you choose the least square or minimum absolute difference values?

Compare the differences in variation among each body part.

How would you compare these differences to see whether the different parts have different amounts of variability?

Find the correlation between body length (height) and each body part.

Discuss how these correlations support or contradict your initial hypothesis (as recorded before you began your work).

Assumptions

Body Part		Expected Correlation (circle one)
Body	+		0	—
Thumb (terminal joint)	+		0	—
3rd Finger (proximal joint)	+		0	—
Forearm (ulna)	+		0	—
Foot	+		0	—
Biceps (circumference)	+		0	—

Data (all measurements in cm estimated to the nearest tenth)

Body Part	Body (height)	Thumb (terminal joint)	3rd Finger (proximal joint)	Forearm (ulna, length)	Foot (length)	Longest toe (length)	Biceps (circum-ference)
You
Classmate
Classmate
Classmate
Classmate
Classmate
Classmate
Classmate
Classmate
Classmate
Classmate
Classmate
Classmate
Classmate
Classmate
Classmate
Classmate
Classmate
Classmate
Classmate
Classmate
Classmate
Classmate
Classmate
Classmate
Classmate
N =