Sampling
Background Information
It is the job of researchers to collect data. Often, data are used to determine it two groups differ. Sometimes, data is used to describe a population. Often the entire population is too large to be completely counted, so samples must be used.
In this activity, students will experience several common sampling techniques: penny tossing, marble selection, and number combinations.
Sampling
Student Activities
1. Penny tossing
Introduction: Each penny has two sides: a head and a tail. If a penny is tossed an even number times, the odds are close that the same number of heads and tails will come up.
a. Toss a penny
What comes up- heads or tails? _____
Record the result on this chart.
Does this result indicate the proportion of heads to tails?
yes_____ no_____b. Toss the penny nine more times.
Record the results on this chart.
Do the nine tosses indicate more clearly the proportion of heads to tails?
yes____ no ____c. Continue tossing the penny 40 more times for a total of 50 tosses.
Record the results on this chart.
How many heads and how many tails come up in the 50 tosses?
heads:_____ tails:_____Which set : one toss, 10 tosses, or 50 tosses, indicates least clearly the proportion of heads to tails?
tosses:_____What is the proportion of heads to tails?
heads:_____% tails:_____%Which set indicates most clearly the proportion of heads to tails?
10 tosses:_____ 50 tosses:_____d. Record on this chart:
Toss no. 1 2 3 4 5 6 7 8 9 10 Heads/Tails Toss no. 11 12 13 14 15 16 17 18 19 20 Heads/Tails Toss no. 21 22 23 24 25 26 27 28 29 30 Heads/Tails Toss no. 31 32 33 34 35 36 37 38 39 40 Heads/Tails Toss no. 41 42 43 44 45 46 47 48 49 50 Heads/Tails
2. Marble selection
For this activity, use items of two colors and the same size: They could be marbles, poker chips, etc. The word "marbles" will be used to refer to the items you choose.
If 20 marbles- 10 white and 10 black- are in a jar, and are selected one at a time, as more marbles are selected, the greater the odds are that the same number of white and black marbles will be chosen, until eventually 10 of each are chosen.
a. Without looking into the jar, select one marble.
- Which color did you select? White or black?_____
Record the result on the chart.- Does this result indicate the proportion of while to black marbles?_____
b. Select four more marbles. Record the results on the chart.
- Do the four selections indicate more clearly the proportion of heads to tails?_____
c. Continue selecting marbles for a total of 20. Record the results on the chart.
- How many white and how many black marbles are chosen in the 20 selections?
white:_____ black:_____- Which set: one marble, four marbles, or 20 marbles, indicates least clearly the proportion of white to black marbles?_____
- Which set indicates most clearly the proportion of white to black marbles?_____
- How many white and how many black marbles are chosen?
white:_____ black:_____
Record on this chart:
| Selection no. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Color | ||||||||||
| Selection no. | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| Color |
3. Number combinations
Information: Random sampling allows the possibility of any number being selected each time. After a number is selected , it must be placed back with the other numbers.
a. There is a set of three numbers: 1, 2, and 3.
The sum of the numbers is 6.
The mean (average) is 2.b. Three cards, each with one of the numbers, is placed in a jar.
c. If two of the numbered cards were chosen at random which two would they be?
- Select a card with a number; record the number.
Place the card back into the jar.
Select a second number; record the second number.
First selection:_____
Second selection:_____
Sum the numbers:_____+_____=_____
Find the mean of the two numbers_____/2=_____
How close to the actual mean was your calculated mean?_____
(example: our first selection of two numbers was: 2+3=5/2=2.5)- Again, select a card with a number; record the number.
Place the card back into the jar.
Select a second number; record the second number.
First selection:_____
Second selection:_____
Sum the numbers:_____+_____=_____
Find the mean of the two numbers_____/2=_____
How close to the actual mean was your calculated mean?_____
(example: our second selection of two numbers:1+2=3/2=1.5)- What is the mean (average) of the two means?_____
(Our overall mean was: 2.5+1.5=4/2=2)- If you were to continue to randomly select two numbers, how close to the actual population mean would your sample mean of means be?
The sample mean of means would probably be:_____.
Charting the Results:
Create a chart using all possible combinations of two numbers randomly selected from the set of 1, 2, and 3.
There are nine sets.
This is an example of a chart:
| First number | Second number | Sums | Mean(#1+#2/2) |
| 1 | 1 | 2 | 1 |
| 1 | 2 | 3 | 1.5 |
| 1 | 3 | 4 | 2 |
| 2 | 1 | 3 | 1.5 |
| 2 | 2 | 4 | 2 |
| 2 | 3 | 5 | 2.5 |
| 3 | 1 | 4 | 2 |
| 3 | 2 | 5 | 2.5 |
| 3 | 3 | 6 | 3 |
Total of means: 18
Mean of means: 2
Chart Using Five Numbers:
Repeat this process using another set of five numbers.
a. Select several pairs of numbers and find their sums and means.
Calculate the mean of meansb. Following the format shown for 3 numbers, fill in this chart to show all possible combinations of two numbers randomly selected from the set. There will be 25 sets.
First number Second number Sums Means (#1 + #2 / 2) _____________ _____________ _____________ _________________ _____________ _____________ _____________ _________________ _____________ _____________ _____________ _________________ _____________ _____________ _____________ _________________ _____________ _____________ _____________ _________________ _____________ _____________ _____________ _________________ _____________ _____________ _____________ _________________ _____________ _____________ _____________ _________________ _____________ _____________ _____________ _________________ _____________ _____________ _____________ _________________ _____________ _____________ _____________ _________________ _____________ _____________ _____________ _________________ _____________ _____________ _____________ _________________ _____________ _____________ _____________ _________________ _____________ _____________ _____________ _________________ _____________ _____________ _____________ _________________ _____________ _____________ _____________ _________________ _____________ _____________ _____________ _________________ _____________ _____________ _____________ _________________ _____________ _____________ _____________ _________________ _____________ _____________ _____________ _________________ _____________ _____________ _____________ _________________ _____________ _____________ _____________ _________________ _____________ _____________ _____________ _________________ _____________ _____________ _____________ _________________
Total of means (Sum the 25 means):_____
Mean of means (Divide the sum of means by 25):_____
4. Summary of sampling activities:
What do these sampling activities demonstrate:
Sampling
Teachers Notes
1. Penny tossing
a. No. It indicates that the penny has either heads or tails, not both.
b. Yes. There should be several heads and several tails.
c. The number should be close to 25 for each.
One toss indicates least clearly the proportion of heads to tails.
50 tosses indicates most clearly the proportion of heads to tails.
2. Marble selection
a. No. It indicates that only one color of marble is in the jar.
b. Yes. There should be several while and several black marbles.
c. There will be 50 white and 50 black.
One marble indicates least clearly the proportion of while to black marbles.
100 tosses indicates most clearly the proportion of while to black marbles.
3. Number combinations
d. Our first selection of two numbers was: 2 + 3=5 / 2=2.5.
Our second selection of two numbers was: 1 + 2=3 / 2=1.5.
Our mean (average) of the two means was: 2.5 + 1.5=4 / 2=2.
This is the actual mean of the three numbers.
Even if it were not the actual mean, it would be closer to the mean than just the first selection of two numbers.
The mean of means should be closer to the actual mean than the mean of any two randomly selected numbers.
4. Summary of sampling activities
These sampling activities demonstrate that as more samples are chosen, they will be closer to the actual population.
Interdisciplinary Approach: Earth Science and Math.