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EDIN 650 Research Methods
Dr. Robert Slotnick
Hand-out #8
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Gay and Airasian, Text, Ch 14 Descriptive Statistics and Ch 15: Inferential Statistics
Review: See Chapter 5.
Nominal Scales
Ordinal Scales
Interval Scales
Ratio Scales
Ch 14: Types of Descriptive Statistics
First step is to describe or summarize data; score all tests reliably
and accurately.
Tabulation and Coding Procedures, p. 434
Organize scoring of all instruments. Prepare for data analysis.
Enter into computer spreadsheet. What statistical test would you
use? See Table 12.1, p. 435 -- it shows Hypothetical Results
of a Study Based on a 2 x 2 Factorial Design; the main IV is Method A and
Method B; the secondary IV is High or Low Aptitude.
The first step in coding data is to give is to give each participant an ID number. If there are 50 Ss, they should be numbereed from 01 to 50. Next, nominal or categorical data include variables such as gender, group membership, and college level. Male and female may be coded as 0 and 1; each category of another variable may also be scored 0, 1, 2, and 3. See p. 435.
Types of Descriptive Statistics
The first step is to describe the data using descriptive statistics.
(When using a sample it is called statistics and when using the population
it is called a parameter.)
Graphing Data, Ch 14, P. 413
The most frequent method of graphing data is to construct a frequency
polygon. List all scores and tabulate how many participants received
each score. Table 12.2 and Figure 12.1 p. 438. Copy the data
from Table 12.2 into a spreadsheet and produce a frequency polygon and
pie chart. It should look like the one shown in figure 12.1.
Looking at the chart what information can you see about the distribution?
Estimate the mean score for the class. Estimate the standard deviation
for the class. Chart a frequency distribution from one of your own
classes. If not possible make up a distribution of about 30 students
and chart that as well. Compare to the frequency polygon in the text.
When using StatPak you must limit your data entries to 30. You can
use Excel for any number and can calculate mean and standard deviation.
Open Excel and enter data and calculate.
Now, let us look as some basic descriptive statistics.
Major types of descriptive statistics are measures of central tendency
-- mean, median, mode
measures of variability -- range, quartile deviation, (75th and 25th
percentiles), variance and standard deviation measures of relative position
-- percentiles, standard scores, z scores and measures of relationship
-- Pearson r (quantitative) and Spearman rho (ranks)
Measures of Central Tendency -- allows researcher to characterize a set of scores or data with a single number. The average or typical score stands for all the scores. The three most frequently used measures of central tendency are the mode, the median, and the mean. The Mode is used to describe nominal data. It is the score attained by the most participants. It is the score which occurs most frequently. A set of scores may have 2 or more modes; it is an unstable measure of central tendency. The Median -- is the 50% point of a distribution of scores. Half the scores are higher and half are lower. For example, for the scores 75, 80, 82, 83, 87, what is the median. 82 is the median. What is the median for the scores, 21, 23, 24, 25, 26, 30. The median is 24.5. When there is an even number of scores you must find the midpoint between the two middle scores. The median is most appropriate when the data represent an ordinal scale. It focuses on the middle scores which is both an advantage and a problem. Different sets of scores may have the same median. The Mean is the arithmetic average of the scores and is the most frequently used measure of central tendency. It is calculated by summing the scores and dividing by the number of scores. It can be used with interval and ratio scales. It is the most precise measure in general, but it can be misleading when there are one or two extreme scores in a distribution. See example in text, p. 440.
Measures of Variabiltiy tell the complementary side of a distribution of scores to the measures of central tendency. The mean describes how similar scores are, the sd descibes how different (or dispersed) a set of scores are. Consider the following sets of data.
Set A 79 79
79 80 81 81
81
Set B 50 60
70 80 90 100
110
The mean of both sets of scores is 80 and the median is 80, but set A is very different from set B. In set A the scores are clustered around the mean, in set B they are dispersed. There is much more variability in set B. We need a meaure of variability. The Range is simply the difference between the highest and the lowest scores. It is useful for nominal or categorical data. The Quartile Deviation is obtained by subtracting the 25th percentile from the 75th percentile and dividing by 2. If the resulting number is small the distribution is close together, and vice versa. Variance and Standard Deviation measures the variation of a set of scores from the mean. The SD is the square root of the variance and is the statistic that is used when the mean is also used. The mean and the sd give you a good picture of the set of scores. If a set of scores match a bell shaped curve (normal curve) then the mean and plus or minus 3 sd encompass more than 99% of all scores.
Normal Curve (p. 443-446) refers to a distribution of scores (of any data) that follows a bell shaped curve. Such a curve shows that most scores are in the middle with fewer and fewer scores as you move away from the middle in either direction. Such a distribution also has certain mathematical properties that make it very interesting to researchers. For example, when a teacher marks on the curve that means that most scores are C, with smaller percentage receiving B and D, and a still smaller percentage receiving A and F.
Normal Distributions
With a normal distribution, 50% of the scores are above the mean
and 50% are below.
The mean, median, mode are the same.
Most scores are near the mean, and the farther from the mean
a score is, the fewer the number of students with that score.
The same number, or percentage of scores is between the mean and plus one standard deviation as is between the mean and minus one standard deviation, and similarly for plus or minus 2sd and 3sd. See Fig 12.2 for characteristics of the normal curve. The mean is 0 and the sd = 1. Therefore, 34% of the cases = 1 sd from the mean and 68% of the cases = +/- 1sd. Plus or minus 2sd = about 95% of the cases. 2.5sd from the mean = about 99% of the cases. The same relationship holds for many different kinds of measurement. Notice all the different measures under the currve. Below the row of sd is a row of percentages. As you move from left to right, the cumulative percentage of scores which fall below each point is indicated.
Measures of Relative Position
indicate where a score is relative to other scores in the distribution.
It allows you to compare students scores on different tests.
Percentile Ranks
indicates the percentage of scores that fall below a given score.
If a score of 65 corresponds to a percentile rank of 80, the 80th percentile,
this means that 80% of the socres in the distribution are lower than 65.
Percentiles are appropriate for data representing an ordinal scale, though
they can be used for interval data as well.
Standard Scores ... is a derived score that tells how far a raw score is from the mean, in terms of standard deviation units. If the distribution represents an interval or a ratio scale, you can use a standard score measure. Standard scores allow scores from different tests to be compared on a common scale and, unlike percentiles can perform mathematical operations on them.
z Scores
express how far a score is from the mean in terms of standard
deviation units. A score which is on the mean corresponds to a z
of 0; a score which is exactly 1 sd above the mean corresponds to a z of
+1.00; a score which is 2 sd above the mean =s a z of +2.00; scores
below the mean are expressed in negative numbers. Once you know the
mean and sd of a distribution, you can calculate the z score and tell the
relative position of each score and you can compare different tests.
The formula is: z = X - Xm/sd. See p. 450
The major advantage of z scores is that they allow scores from
different tests to be compared. Review Table A.3 in the Appendix
to see the distributions for the z scores. For any z score we can
also read what percentage of cases are included so far.
T scores
are z scores expressed in positive numbers. To transform
a z score to a T score, you simply multiply the z score by 10 and add 50.
This eliminates the negative numbers. The T score distribution has
a mean of 50 and a sd of 10.
Something to think about: How does an individual stand
relative to his classmates, who received a score of 83 on the first test
and a score of 90 on the second test? Which raw score yields the
higher z score? What would be the likely standing of the student
in the class for each score? On which test did he do better?
The best way to answer this question is to calculate the z or T scores
for the students. A T or a z score will tell you the relative position
of all students on a test, etc. (z = 90 -80 = 10/10 = 1 compared
to 83 - 80 = 3/1 = 3). Explain why the score of 83 is relatively
higher than the score of 90.
Reference Tables, Appendix A, p.601
Table A.1 Ten Thousand Random Numbers
Table A.2 Values of the Correlation Coefficient for Different
Levels of Significance
Table A.3 Standard Normal Curve Areas (z Scores)
Table A.4 Distribution of t
Table A.5 Distribution of F
Table A.6 Distribution of Chi Square
Stanines. Standard Nines. Stanines are standard scores that divide a distribution inot nine parts. The formula for deriving a stanine is 2z + 5 and round to the nearest whole number. Stanines 2 through 8 each represent 1/2 SD.
Measures of Relationship See Ch 9 for detailed discussion.
What degree of relationship exists between two or more quantifiable
variables. Degree of relationship is expressed as a correlation coefficient.
Two sets of scores (for ex., test 1 and test 2) from a single group of
participants.
The Spearman Rho coefficient is used to correlate data that are ranked. The data represent an ordinal scale. Data variables are ranks.
The Pearson r correlation coefficient is the most appropriate measure when the variables are either interval or ratio and the data variables are both quantitative. Most education tests are interval and we use the r correlation. It is more sensitive than rho and takes into account each score, while rho measures ranks. In both correlations the range of coefficients is +/- 1.00.
Calculation for Interval Data
See pages 452 to 463 for examples of calculating various statistics
by hand and then using StatPak. The mean and sd are calculated by
hand and then with StatPak. Then standard scores are calculated by
hand and then with StatPak. Finally, Pearson's r correlation is calculated
by hand and then by StatPak. The comparisons allow you to see underlying
logic of the various statistics and then allows the computer to calculate
the results less painfully.
Review: Calculation of the mean; the Standard Deviation
-- is equal to the square root of the sum of squares divided by N-1.
Compare hand calculation to StatPak. Review calculation of Standard
Scores and compare to StatPak. Conclude with the calculation of the
correlation coefficient.
Assignment 5: Frequency Distribution, Polygon,
and Chart
Examples Based on Ch 12, Classwork and Homework
The most frequent method of graphing data is to construct a frequency
polygon. Copy the data from Table 12.2 into a spreadsheet and
produce a frequency polygon, bar chart and pie chart. It should look
like the one shown in figure 12.1. How does this distribution compare
to the normal curve? List all scores and tabulate how many
participants received each score. Looking at the chart what information
can you see about the distribution? Estimate the mean score for the
class. Estimate the standard deviation for the class.
Hint: To create a pie chart that resembles the one in the text, on your spreadsheet you must convert the individual scores to intervals. Use exact same data for your spreadsheet as in the text example. Then make chart in any spreadsheet. Make a pie chart, then experiment with other types.
Chart a frequency distribution from one of your own classes. If not possible make up a distribution of about 30 students and chart that as well. Compare to the frequency polygon in the text. When using StatPak you must limit your data entries to 30. You can use Excel for any number and can calculate mean and standard deviation. Open Excel and enter data and calculate.
See example: Sample Frequency data
Assignment 6: Stat Pak Data Set, based on Ch
12. Hand-out #8
1. These are the scores on a math test. What
is the mean and the sd for the class?
Scores: 78, 84, 84, 80, 81, 82, 89, 78, 88, 80, 92, 92,
85, 90, 84, 85, 85, 83, 87, 88.
2. Find the mode, median and mean for the following distribution
of reading scores.
75, 80, 80, 81, 82, 82, 82, 82, 84, 84, 86, 88, 90
3.These are the scores from Group 1 on a 6th grade reading test: 78, 84, 84, 80, 81, 82, 89, 78, 88, 80 ; The following scores are from Group 2 on the same test: 92, 92, 85, 90, 88, 87, 85, 89, 87, 88. Which group scored higher? Conduct a t-test for independent groups. What is the df? What is the p value? Is there a significant difference between the groups?
4. Consider the following test distributions for the same class
in reading and math.
Scores: T1(math) = 55, 60, 65, 65, 70, 75, 80, 80, 85, 90, 90,
95, 85, 90, 95
Scores: T2(reading) = 42, 51, 58, 55, 60, 48, 49, 53, 56, 48, 47, 49, 52, 53, 56
John received a score of 70 in math and a score of 58 in reading. How would you compare the 2 scores? For these two distributions of scores create a distribution of z scores and T scores. On which test did he receive a higher relative standing? Also, find the percentile equivalents of the two scores. How do you find the percentile from the z score?
5. Analyze the following data:
Gr 1: 14, 15,16,17, 18, 19
Gr 2: 10, 11, 12, 14, 16, 17
Gr 3: 21, 22, 23, 24, 25, 26
Identify the correct statistical test and perform the analysis.
What is the F ratio? Do you need to perform a Schefe test?
What are the results?
6. Assume the following data are true and identify the correct statistical
test and analyze:
Females voted 67% for Gore and 33% for Bush
Males voted 46% for Gore and 54% for Bush
Is there a significant relationship between Gender and Choice of Candidate?
What is the statistical test? What are the results? Are
they significant?
In ch 12 you learned different methods of describing a population and in ch 13, the main idea is how to make inferences about a population.
Standard Error (of the mean)
Inferences about populations are based on the behavior of samples. Try to obtain a true random sample for your study. The sample should be a true reflection of the population. If the sample is biased, then the results may be due to the bias and not due to the treatment (independent variable). You certainly cannot generalize your results to the entire population, but it is possible that the results are not true even for your sample.
Likeliness that results based on samples are the results that would have occurred for the entire population. This allows you to generalize from your sample to the entire population. If sample is biased or flawed, then you cannot generalize.
Main Point
If a difference exists the question is whether the difference
could have occurred by chance or if it is a real difference due to the
independent variable.
The Null Hypothesis says that there is no true difference between parameters in the population (no difference between 2 means) and that any difference or relationship found for the samples is the result of sampling error. Any difference you may have found in your study is due to chance and not to the Independent Variable (manipulation).
Tests of Significance -- method used to show whether a real difference exists. Result obtained from the test is compared to the probability on a table of whether such a difference could be obtained by chance. If yes, you accept the null hypothesis; if not, you reject the null hypothesis and accept your hypothesis that a real difference exists between the groups. The test of significance is made at a predetermined probability level. Usually, researchers select 5% or 1% as the cut-off point. That means that such a result could occur by chance 5 times or 1 time in a hundred.
Of course that means when you reject the null hypothesis at the 5% level, you are correct 95% of the time and wrong 5% of the time. It also means when you accept the null hypothesis as true, sometimes it will be incorrect and should be rejected. The first is called a Type I error and the second is called a Type II error. See page 475.
Significance level of 5% is actually at the 2 sd difference between the groups. Significance at the 1% level is actually at the 3 sd difference between the two groups. See fig 13.2, p. 478.
Types of Significance Tests
The t test is used to determine whether two means are significantly
different at a selected probability level. When sample size is small,
the difference needed is large. When sample size is large, the difference
needed is much smaller. Similarly, when the significance level becomes
smaller from .1 to .05 to .01, etc. the difference needed becomes larger.
The table in the appendix shows the value needed to reject the null hypothesis
at each significance level for group size. The text shows you how
the t test is calculated by hand and with StatPak.
Simple Analysis of Variance (ANOVA)
The method to use when there are more than 2 groups. If you are
comparing 3 groups, for example, three different methods of teaching reading,
then the ANOVA is used. ANOVA tries to answer the question whether
the differences among the mean values are due to chance or are significantly
different from each other. An F ratio is computed and the value is
compared to the values in table A5. ANOVA is simpler and more efficient
to compute than a whole bunch of separate t-tests.
Multiple Comparisons
If the F ratio is significant, we must find out which comparison is
significant. A significant F ratio tells us that there is at least
one significant difference among the means. Now, you must determine
which of the means are significant. The best way to determine this
is with a multiple comparisons test. A regular t test is not acceptable
because multiplying th number of t tests increases the chance of significance,
or a type I error (rejecting when you should accept). The Scheffe
test is appropriate for making any and all possible comparisons involving
a set of means.
Factorial Analysis of Variance
When you are measuring 2 or more independent variables and the interactions
between them, the appropriate statistical analysis is a factorial analysis
of variance. The factorial analysis provides a separate F ratio for
each independent variable and for each interaction.
Chi Square is called a nonparametric test because the variables
compared are categorical not quantitative. It compares percentages
not actual scores as in ANOVA. Comparisons may be made between Male
and Female or tall or short. A chi sq test compares the proportions
observed to the proportions expected. You may compare one dimensional
chi sq or two or more dimensional chi sq. See Page 503. See
examples of chi sq calculated by hand and compared to StatPak for one dimensional
and two dimensional situations on pages 504 to 508. Figure 13.5 on
p. 509 is a good example of different tests of significance, their purpose
and types of variables compared.