Even More About z-Scores

November 22, 2008 by  
Filed under Even More

There are 5 primary applications of z-scores.

First, z-scores can be used for describing the location of an individual score. If your score is at the mean, your z score equals zero. If z = 1, you are one standard deviation above the mean. If z = -2, you are two standard deviations below the mean. If z = 1.27, you score is a bit more than one and 1/4th standard deviations above the mean.

What is the z-score for a raw score of 104, a mean of 110 and a standard deviation of 12? 104-110 equals -6; -6 divided by 12 equals -.5. The raw score of 104 is one-half a standard deviation below the mean.

Second, raw scores can be evaluated in relation to some set z-score standard; a cutoff score. For example, all of the scores above a cutoff z-score of 1.65 could be accepted. In this case, z-scores provide a convenient way of describing a frequency distribution regardless of what variable is being measured.

Each z score’s location in a distribution is associated with an area under the curve. A z of 0 is at the 50th percentile and indicates that 50% of the scores are below that point. A z score of 2 is associated with the 98th percentile. If we wanted to select the top 2% of the individuals taking a musical ability test, we would want those who had a z score of 2 or higher. Z scores allow us to compare an individual to a standard regardless of whether the test had a mean of 60 or 124.

Most statistics textbooks have a table that shows the percentage of scores at any given point of a normal distribution. You can begin with a z score and find an area or begin with an area and find the corresponding z score. Areas are listed as decimals: .5000 instead of 50%. In order to save space, tables only list positive values are shown. The tables also assume you know that 50% of the scores fall below the mean and 50% above the mean. The table usually has 3 columns: the z score, the area between the mean and z, and the area beyond z.

The area between the mean and z is the percentage of scores located between z and the mean. A z of 0 has an area between the mean and z of 0 and the area beyond (the area toward the end of the distribution) as .5000. Although there are no negatives, notice that a z score of -0 would also have an area beyond (toward the low end of the distribution) of .5000.

A z score of .1, for example, has an area between the mean and z of .0398. That is, 3.98% of the scores fall within this area. And the third column shows that the area beyond (toward the positive end of the distribution) is .4602. If the z has -.1, the area from the mean down to that point would account for 3.98% of the scores and the area beyond (toward the negative end of the distribution) would be .4602.

Areas under the curve can be combined. For example, to calculate the percentile of a z of .1, the area between the mean and z (.0398) is added to the area below z (which you know to be .5000). So the total percentage of scores below a z of .1 is 53.98 (that is, .0398 plus .5000). A z score of .1 is at the 53.98th percentile.

Third, an entire variable can be converted to z-scores. This process of converting raw scores to z-scores is called standardizing and the resulting distribution of z-scores is a normalized or standardized distribution. A standardized test, then, is one whose scores have been converted from raw scores to z-scores. The resultant distribution always has a mean of 0 and a standard deviation of 1.

Standardizing a distribution gets rid of the rough edges of reality. If you’ve created a nifty new test of artistic sensitivity, the mean might be 123.73 and the standard deviation might be 23.2391. Interpreting these results and communicating them to others would be easier if the distribution was smooth and conformed exactly to the shape of a normal distribution. Converting each score on your artistic sensitivity test to a z score, converts the raw distribution’s bumps and nicks into a smooth normal distribution with a mean of 0 and a standard deviation of 1. Z scores make life prettier.

Fourth, once converted to a standardized distribution, the variable can be linearly transformed to have any mean and standard deviation desired. By reversing the process, z-scores are converted back to raw score by multiplying each by the desired standard deviation and add the desired mean. Most intelligence tests have a mean of 100 and a standard deviation of 15 or 16. But these numbers didn’t magically appear. The original data looked as wobbly as your test of artistic sensitivity. The original distribution was converted to z scores and then the entire distribution was shifted.

To change a normal distribution (a distribution of z scores) to a new distribution, simply multiply by the standard deviation you want and add the mean you want. It’s easy to take a normalized distribution and convert it to a distribution with a mean of 100 and a standard deviation of 20. Begin with the z scores and multiply by 20. A z of 0 (at the mean) is still 0, a z of 1 is 20 and a z of -1 is -20. Now add 100 to each, and the mean becomes 100 and the z of 1 is now 120. The z of -1 becomes 80, because 100 plus -20 equals 80. The resulting distribution will have a mean of 100 and a standard deviation of 20.

Fifth, two distributions with different means and standard deviations can be converted to z-scores and compared. Comparing distributions is possible after each distribution is converted into z’s. The conversion process allows previously incomparable variables to be compared. If a child comes to your school but she old school used a different math ability test, you can estimate her score on your school’s test by converting both to z scores.

If her score was 65 on a test with a mean of 50 and a standard deviation of 10, her z score was 1.5 on the old test (65-50 divided by 10 equals 1.5). If your school’s test has a mean of 80 and a standard deviation of 20, you can estimate her score on your test as being 1.5 standard deviations above the mean; a score of 110 on your test.

 

NOW YOU CHOOSE:
    Day 4: z-Score
    A Bit More About z-Scores
    Even More About z-Scores
    How To Calculate z-Scores
    Practice Problems
    Basic Facts About z-Scores
    Vocabulary
    Formulas For z-Scores
    Quiz 4
    Summary

Even More About Standard Deviation

November 22, 2008 by  
Filed under Even More

 Like the other four measures of dispersion, the standard deviation gets smaller as the scores get more homogeneous, and larger the more heterogeneous they become. A small standard deviation indicates the scores are quite similar to the mean (closer to the peak); a large standard deviation says the score vary from the mean.

SQUARE ROOT

 

This measure of dispersion is calculated by taking the square-root of variance. Regardless of whether you used N or N-1 to calculate variance, standard deviation is the square-root of variance. If variance is 7.22, the standard deviation is 2.69. If variance is 8.67, the standard deviation equals 2.94.

Technically, the square-root of a population variance is called sigma and the square-root of a sample variance is called the standard deviation. As a general rule, population measures use Greek symbols and sample parameters use English letters. Since we tend to use large samples, we’ll focus on the standard deviation.

STANDARD

The standard deviation is “standard” in the sense that it takes steps of equal distance from the mean. Think of it as standing at the mean and taking 3 steps in one direction. If doesn’t matter if you step toward the high end or the low end. It only takes three steps to get from the mean to the end of a distribution. If you start at the mean and go toward the positive end, you’re there in 3 steps; and it’s 3 steps from the mean to the lowest end of the distribution. So the entire distribution is comprised of 6 steps (3 positive steps and 3 negative steps).

STEPS (Deviations From The Mean)

 

Each of these steps is the equal in distance but accounts for a different amount of people. The normal curve is like a mountain. If you’re standing on top of the mountain, your first step is always your largest. In a frequency distribution of a normally distributed variable, your first step accounts for the most people. Because most scores are close to the mean, most scores fall within plus or minus one standard deviation from the mean.

In fact, that’s our definition of normal. Normal is being close to the mean. Normal musical ability is scoring at the mean plus or minus one standard deviation. Normal basketball throwing is at the mean, plus or minus one standard deviation.

In a normally distributed variable, the percentages are consistent, regardless of what is being measured. Starting from the mean, the first step accounts for just over 34% of the scores. The next steps has 14% and the last step has 2%. Since normal frequency distributions are symmetrical, the percentages work on either side of the man. So the entire distribution looks like this: 

 

 

 

The majority of scores fall at the mean, plus and minus one standard deviation. Normal, then, includes 68% of the scores. There are 68% of the scores from one standard deviation below the mean to one standard deviation above the mean.

NOW YOU CHOOSE:
    Day 3: Dispersion

   
A Bit More About Dispersion
    Even More About Dispersion
        Range
        MAD
        Sum of Squares
        Variance
        Standard Deviation
    How To Calculate
        Range
        MAD
        Sum of Squares
        Variance
        Standard Deviation
    Formulas For Dispersion
    Practice Problems
    More Practice Problems
    Basic Facts About Dispersion
    Vocabulary
    Quiz 3
    Summary

     

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

  

 

 

 

 

  

 

   

 

 

 

 

 

 

 

 

  

 

 

 

 

  

 

   

 

 

 

 

  

 

   

 

    

 

 

 

 

 

 

 

 

  

 

 

 

 

  

 

   

 

 

 

 

  

 

   

 

    

 

 

 

 

  

 

   

 

    

 

     

 

 

 

 

 

 

 

 

  

 

 

 

 

  

 

   

 

 

 

 

  

 

   

 

    

 

 

 

 

  

 

   

 

    

 

     

 

 

 

 

  

 

   

 

    

 

     

 

      

 

 

 

 

 

 

 

 

  

 

 

 

 

  

 

   

 

 

 

 

  

 

   

 

    

 

 

 

 

  

 

   

 

    

 

     

 

 

 

 

  

 

   

 

    

 

     

 

      

 

 

 

 

  

 

   

 

    

 

     

 

      

 

       

 

 

 

 

 

 

 

 

  

 

 

 

 

  

 

   

 

 

 

 

  

 

   

 

    

 

 

 

 

  

 

   

 

    

 

     

 

 

 

 

  

 

   

 

    

 

     

 

      

 

 

 

 

  

 

   

 

    

 

     

 

      

 

       

 

 

 

 

  

 

   

 

    

 

     

 

      

 

       

 

        

 

    

Even More About Measurement 2

November 12, 2008 by  
Filed under Even More

3. Who Is Predicting Whom?

 In general, we believe that most variables are continuous. People aren’t just smart and stupid, they vary on a continuous scale of intelligence. People are not just rich and poor, their earnings are better described by a continuous variable. Even drug abuse can be considered on a continuous scale (amount of drugs consumed).

Although the underlying variable is continuous, how the questions are worded can make the data appear discrete. Although years of school is a continuous variable, the question “have you ever gone to school?” would result in non-continuous (discrete) data. “Are you employed?” produces discrete information, but the number of days worked is continuous. It is possible to study a continuous variable in a way which makes it look discrete.

Continuous data, then, is a factor which can describe people on a large scale with small steps; discrete data is a continuous variable chopped up into parts (high, medium, low; fast, slow). A discrete variable with only two levels (e.g., yes, no) has its own name: dichotomous.

Traditionally, a differentiation is made between independent and dependent variables. It is a characterization based on locus of control. A dependent variable depends on the performance of the subjects. It is anything that we measure, observe or record. A dependent variable is an outcome. In contrast, an independent variable is independent of the subjects’ control. It is something the researcher selects, manipulates or induces.

The distinction is clearest in a traditional experiment: an independent variable is manipulated and a dependent variable is measured. Such a structure provides confidence in making inferences of causation. You stomp on a foot, the person says “ouch.” You don’t stomp on the foot and the person says nothing. The clear inference is that stomping on a foot causes a person to say “ouch.”

Notice that the independent variable is a discrete variable: stomp or not-stomp. It is not measured in continuous increments of pressure but is either there or absent. A variation of this theme is to select high, medium and low levels of an independent variable but, again, the independent variable is a discrete variable that is manipulated to see what impact it has on a continuous dependent variable.

In many areas of research, variables can not be directly manipulated, if at all. It would be ridiculous and unethical to assign children to abusive and non-abusive environments to see what impact the independent variable (abuse) has on the dependent variable (self-esteem, for instance). Consequently, the independence of many “independent variables” is in question.

Also, the more complicated models of human behavior include many variables, each impacting and being impacted upon by others. These experimental designs do not lend themselves to the independent-dependent variable distinction. Consequently, there is much to recommend the replacement of independent-dependent variables with the designation of predictor-criterion.

As an alternative to the independent-dependent variable characterization, the predictor-criterion designation provides more flexibility and more accurately depicts the relationships between model components.

Predictor-criterion is more flexible because it includes discrete and continuous variables. Although a discrete predictor (stomp or don’t stomp) is good, a continuous predictor would give more information about the amount of pressure needed before you said “ouch.” Also, when it is impossible to manipulate a situation (such as height, gender, or personality type), the term “independent” doesn’t aptly describe the variable. Predictors can be discrete (like a traditional independent variable) or continuous (like a correlation or regression). The predictor-criterion distinction also is a better description of the relationship between the variables. When subjects cannot be randomly assigned to treatments, the independence of variables is in question. It is clearer to note that a particular variable is being used as a predictor of another.

This approach accommodates both traditional experimental designs and complex correlational and causal modeling designs. In addition to simple discrete predictor and continuous criterion, the same nomenclature can be used for continuous predictors, moderator variables (ones that influence only part of a model), intervening variables (variables stuck between a predictor and a criterion) and suppressor variables (variables that filter out noise).

It is important to note that in an actual research study, any variable can be a predictor or a criterion. Annual income, level of education, self-esteem, intelligence-any could be used as a predictor of another. And each could be a criterion. Since the choice is arbitrary, the choice of model components and the hypothesized interrelationships should be determined by the theory being studied. The type of relationships between model components is determined by our theoretical questions

4. Who Are You Going To Study? Sometimes researchers want to study an entire population: the total number of subjects in a particular area of interest. As the focus of interest changes, the size of the population being studying changes. If you’re only interesting in what happens to you, the population of interest is 1.

Although we think of population as the number of people in a city or country, in research, a population is any group of interest. It can the number of people in a family, the number of dogs in a town, or the number of lights on a Christmas tree. Sometimes the population of interest is too large to measure directly. It is usually not convenient to talk to all of the people in a county or inspect all of the paper clips made daily. When the population is too large, a sample is chosen.

A selected part of a larger group is called a sample. Any group can be thought of as both a collection of smaller groups (a population) and a sample of a larger group. The students in Ms. Mendoza’s class are a population to her, a sample of all the 4th graders in the school, a sample of all of the students in the school district, etc.

Obviously, how a sample is chosen determines how well it represents the population. If the first 10 children who enter the class are selected, Ms. Mendoza might have excluded those who rode on the bus (if it ran late that day). A common practice is random selection. Each subject is selected at random from the convenient pool of subjects. Subjects are randomly selected from those students taking introductory psychology classes who want to participant in a study.

An alternative method is called stratification. Like rock walls, groups of people are composed of segments or layers. When there are certain subgroup comparisons you want to make (male-female, rich-poor or tall-medium-short, for example), subjects are randomly selected from within the categories. First, categories of interest are selected. Then, subjects are randomly selected within each category.

However, the best way to pick a sample is random sampling. If everyone in the population of interest has an equal opportunity to be selected, the sample is unlikely to be biased in favor of any particular subgroup. This is very seldom done…for very good reasons. Although researchers want to draw conclusions about the people in general, each person does not have an equal chance of being selected. People living in rural areas-and the disabled, elderly and very young-are generally not included in studies. When interpreting research results, it is important to remember the limitations of sampling.

 

NOW YOU CHOOSE:
   
Day 1: Measurement
    
A Bit More About Measurement
    Even More About Measurement 1
    Even More About Measurement 2
    Even More About Measurement 3
    Basic Facts About Measurement
    Vocabulary
    Quiz 1
    Summary

Even More About Measurement 3

November 12, 2008 by  
Filed under Even More

5. What Do The Numbers Mean? Case studies don’t use numbers. And N=1 studies limit the use of numbers to counting. In contrast, most other approaches to research use numbers to measure and describe groups of people. But what meaning do the numbers have?

Obviously, variables do not always use numbers in the same way. You might want to find the average age of a group of people but it’s unlikely, for example, that you will want to calculate the average ID number. You know intuitively that averaging ID numbers, room numbers, or Social Security numbers isn’t very useful. Such numbers aren’t used for their numerical value but simply as names.

A number which substitutes for a name makes no mathematical assumptions. A marathon runner with a high number of his back doesn’t necessarily run faster than one with a small number. The numbers are only used to be able to tell the difference between contestants. Such numbers are at the lowest level of assumption, and are said to be at a nominal level of measurement.

There are four levels of assumptions which can be made about numbers. At the nominal scale, we assume that the numbers we obtain can be used to simply distinguish between entities. The numbers on the jerseys of football players, for example, help us to distinguish between players. It makes no sense to add these numbers together, or find their average; each number is used as a name (nom).

The exuberance of this runner is not coming in 8th place. Although I’d be thrilled with that performance, his smile is for winning the race. The number 8 is just a nominal marker. It has no mathematical value.

In contrast, the second level of measurement, ordinal, makes two assumptions about its numbers. An ordinal scale distinguishes between members plus places them in order. Ranking children from tallest to shortest is an ordinal measurement. Winners of a race can be placed in order of 1, 2, and 3 (first, second, and third) but it would be silly to find the average of these numbers. An ordinal scale is like a footrace in a snowstorm: it can tell who came in first but it can’t tell how far apart the runners are.

An interval scale includes both of the previous assumptions plus the assumption that the distances between numbers (intervals) are equal. The distance between a score of 8 and a score of 9 on a spelling test is the same distance apart as 3 and 4. Using an interval scale, we could tell the difference between players, find out who came in first, and determine by how much our spelling star won.

Notice, that an interval scale assumes equal intervals. In the case of a test, equal intervals means that each item is equally difficult. When the steps are not equal, the scale is ordinal. Consequently, a lot of teacher-made tests look as though they are based on an interval scale but are in fact making ordinal measurements.

The final level of measurement is ratio. A ratio scale includes the previous three assumptions and adds an absolute zero. Because of their absolute zeros, ratio scales have a unique characteristic: they can be used to make ratio comparisons. We can say that a task took twice as long (a ratio of 2 to 1), or that an object weighs a third as much (a ratio of 1 to 3). Our judgments can be described in relation to each other. We can’t do that with nominal, ordinal or interval scales.

A 0 on a spelling test doesn’t mean that the person cannot spell anything at all, only that those selected words couldn’t be spelled. The zero is not absolute. Similarly, a 0 on a Fahrenheit thermometer doesn’t indicate a total lack of heat (if it did we couldn’t have minus degrees). In contrast, time, distance, and weight are all ratio scales. A 0 on these scales indicates the total absence of that factor.

There are two problems with ratio scales. First, ratio scales are very rare. We often use interval scales (e.g., intelligence scales, reading tests, personality inventories) or ordinal scales (e.g., rating scales), but do not often use ratio scales.

The second problem is that measurement levels often are ignored. It is common for executives, teachers and others to treat ordinal and interval data as if they were on a ratio scale. Rating scales (1 to 5, 1 to 7, 1 to 10) are ordinal in nature. This is important to understand because some people make the mistake of saying that Group A did twice as well as Group B in the last survey.

When our measurements do not meet the assumptions of a ratio scale, we cannot say that a person with an IQ of 140 is twice as smart as a person with an IQ of 70. Nor can we say that a person who scores 0 on our extroversion scale is not extraverted. These are interval scales.

The way we measure determines the strength of the conclusions we draw. If we label horses as “jumpers” and “non-jumpers,” we have not made any assumptions about which is better, only that they are different. This is a nominal scale. Similarly, if we differentiate between managers and engineers at a nominal level, we make no assumptions concerning which status is best.

At an ordinal level, we could rate horses on their jumping ability or personnel on their sales ability. We could use a scale of 1 to 5, for example. Notice, that we could use a two-level scale: thumbs up, thumbs down. The only difference between a two-level ordinal scale and the nominal scale mentioned above is in the assumptions. If we assume that jumpers are better to have than non-jumpers or that sellers are better than non-sellers, the underlying scale is not nominal but ordinal.

Prejudice is a good example of this. Distinguishing between Asians and Whites, north and south, or tall and short is a nominal description. Yet, if underlying our distinctions there is an assumption of one being better than another, we have moved to a different level of measurement.

It should be clear that the number of spots on an ordinal scale is arbitrary. We are still at an ordinal level when rating on a 10-point scale, a 50-point scale or a 87-point scale. It is the underlying assumptions which determine an item’s level of measurement.

To move up to an interval scale in our horse testing, we could set up a course with obstructions for the horses to jump. Again, the number of hurtles included in the course is arbitrary and does not affect the level of its measurement. And, it should also be clear that a score of zero on our hypothetical course does not mean that a given horse cannot jump at all. We may have made all of the jumps too high for any horse to successfully clear.

If we measured how fast each horse ran the course, or how high each one jumped, the measurements would be on a ratio scale. Then, and only then, could we say that one horse jumped twice as high or ran half as fast.

In social science much of our data is ordinal. When we build a test, we usually don’t make each item of equal difficulty (one of the assumptions for an interval scale). Consequently, our measurements are more like rating scales than precision scientific instruments. Although some of our rating systems are quite complex, the data does not allow us to make fine distinctions between people. We can say one person is more generous, skilled or intelligent than another, but not by how much.

Just as horse-jumping courses usually are composed of items with varying difficulty, items of sales ability differ in difficulty. We do it to save time. With a few items of increasing difficulty, we can distinguish between poor performance and great performance. Without thinking about it, though, we have shifted the underlying level of measurement to an ordinal scale.

This shift is not necessarily bad. It allows us to make gross distinctions with only a few items. But researchers should know which level of measurement they are using. Without such knowledge, they are relying on assumptions which might not be true. We should not fool ourselves into thinking that we are measuring with more precision than is actually present.

Clearly, every level of measurement can be useful. Our tests of increasing difficulty are valuable. We don’t have to measure everything as ratio data. We can use nominal, ordinal, interval and ratio data. All are useful. Levels of measurement are themselves nominal: one level is not better than another.

 

NOW YOU CHOOSE:
   
Day 1: Measurement
    
A Bit More About Measurement
    Even More About Measurement 1
    Even More About Measurement 2
    Even More About Measurement 3
    Basic Facts About Measurement
    Vocabulary
    Quiz 1
    Summary

Even More About ANOR

November 9, 2008 by  
Filed under Even More

A correlation is a measure of commonality; how much two variables have in common. For the Pearson r, we plotted two continuous variables and looked at the scatterplot of the data. We could see if the trend was generally positive, negative, or had no linear pattern. We then used a regression for predicting. We plotted a regression line through the data as best we can, using the line to make predictions.

An analysis of regression looks at the pattern of data and compares it to the regression line drawn through it. It asks how well the data looks like a straight line. This is a yes-no comparison. We start with the premise that the data doesn’t look like a straight line. We assume that there is no pattern. When we see small variations from a chance pattern, we still don’t accept the model of a straight line. We only change our minds when the pattern is so strong that it is significant.

Our test of significance is a ratio of knowledge. We’re going to compare the variance we understand to the variance that is unexplained. We are going to compare variation between people to the variation within a person’s performance. Later, we will use this procedure to compare differences between experimental groups to variation within each group. That is, we will compare between-subjects variance to error variance (within-subjects variance).

For the present, we can use the same test (Fischer’s F) to test the significance of a regression. Does the data we collected approximate a straight line? To find out, we’re going to divide the variance the two variables share by the variance they don’t share.

To make the process easy, the F test uses a summary table. We just fill in the gaps. The table looks like this:

                                        Sum of Squares       df          mean squares
          SS regression             
          SSerror                       
          SStotal                       

Assuming we’ve already calculated the sum of squares for each variable (X and Y) and the SSxy, filling out the table is really easy. It’s a three step process. First, we find the Sum of Squares for each component. Starting at the bottom row, SStotal equals SSy. No further calculation is necessary to fill in that answer.

 Let’s use an example and follow it through the process. Here’s the data:

                   X               Y
                    2               4
                    5               7
                    3               9
                    6               8
                  11             10
                  12             10

In this example, the SSx is 85.5, the SSy is 26 and the SSxy is 36. The correlation between the two variables equals .76. So in our summary table, we put the SSy as the SStotal, and the table looks like this:

                                       Sum of Squares       df          mean squares
          SS regression             
          SSerror                       
          SStotal                        26

Moving to other two rows, we partition the Sum of Squares (SS) of the regression into explained SS and unexplained SS. Explained SS is simply the SSy multiplied by r2 (which is called the coefficient of determination). The result is the SSregression; it’s the SS we understand (the part the two variables share). In this example, it is 26 times .58, which equals 15.08.

Similarly, the SSerror is the SStotal times the coefficient of nondetermin-ation (1-r2); in this case that would be .42 times 26 = 10.92. Of course you also could subtract the SSregression from the SStotal. Either way will work.

So far, at the end of step 1, the summary table would look like this:

                                     Sum of Squares       df          mean squares
          SS regression             15.08
          SSerror                       10.92
          SStotal                        26

We’ve partitioned the Sum of Squares into the portion explained by the regression (15.20) and the portion that is due to error (10.92). In this context, anything that isn’t explained by the regression line is considered error.

Step 2 is to identify the appropriate degrees of freedom for each Sum of Squares. You’ll recall that variance is SS divided by its degrees of freedom. A single distribution of scores from a population had a df of N, and a distribution from a population has a df of N-1. In this case, we have two variables so finding the appropriate degrees of freedom is a bit different.

The degrees of freedom (df) for Regression is the number of columns minus one (they call it k-1; k for columns?). Since a simple linear regression has only 2 columns, the df for an Analysis of Regression always equals 1. The df for Error is N-k (number of people minus the number of columns). And Total error = N-1. The summary table now looks like this:

We complete Step 2 by dividing through the appropriate degrees of freedom. There are columns in our ANOR, so df for regression equals 1. There are six people in the study (2 scores for each person), so the df for error is N-k (6 minus 2) which equals 4. And the total degrees of freedom is 6-1 (N-1). So our summary table now looks like this:

                                       Sum of Squares        df            mean squares
          SS regression             15.08                  1
          SSerror                       10.92                   4
          SStotal                        26                       5

Step 3 is to to convert Sum of Squares to variance. We divide each SS by its respective degrees of freedom. The resulting variance terms are called mean squares (just to confuse you). I suppose it is a reminder that variance is the average of the squared deviations from a distribution’s mean. But changing the names does make it harder. Here are the results of our 3-step process:

                                         Sum of Squares       df             mean squares
          SS regression             15.08                   1                 15.08
          SSerror                       10.92                    4                   2.73
          SStotal                        26                        5                   5.20

Of course, the mean squares won’t add up like the other columns because we divided by different amounts But the resulting variance terms are appropriate for their respective portions.

To calculate F itself, we divide the mean squares of regression by the mean squares of error. In this example 15.08 is divided by 2.73, and F = 5.52.

In order for the value we calculated to be deemed significant, if must be larger than a standard value for that size of a data set. We compare the F we calculated to the F table at the back of nearly any statistics book. To find the right value, we select the first column (the same value as the df for SSregression). And to find the correct row, we go down to the row labeled 4 (the same value as the df for SSerror). In this case the book value is 7.71. Our F value was 5.52, so we lose.

Well, lose isn’t really the right word but if you think of research as trying to beat the book value, it will help you remember for to make the comparison. To be significant, F has to be equal to or larger than the one in the Critical Values of F table. If F is equal to or larger than the book value, we win. If our F is smaller than the book’s, we lose.

The proper explanation is that F indicates the likelihood that what we see is not due to chance. If our F is smaller than the book, what we see is likely to be due to chance. If our F is equal to or larger than the book, the relationship between variables is likely to be due to something other than chance.

The F test doesn’t tell us what causes what, only whether it is a likely occurrence or not. In this example, there is no significant impact of X on Y. Any apparent causal relationship can be explained by chance.

 

NOW YOU CHOOSE:
    Day 7: Probability
    
Bit More About Probability
    Even More About Probability
    Even More About ANOR
    Calculate ANOR
    Practice Problems
    More Practice Problems
    Word Problems
       
Sim1       Sim2        Sim3
    Basic Facts About Probability
    Vocabulary
    Formulas
    Quiz 7
    Summary

More Mode

November 5, 2008 by  
Filed under Even More

The mode of the following is 5 and 7; it is a bimodal distribution:

9
7
7
7
8
6
5
5
5

The mode of the following is 6 and 2; it is a bimodal distribution:

6
2
6
7
8
6
2
5
2

 

NOW YOU CHOOSE:
   Day 2: Central Tendency
   
A Bit More About Central Tendency
   Even More About Central Tendency
   More Examples
     
More Mean Examples

      More Median Examples
      Median Is Middle Of Distribution
      More Mode Examples
   Impact of Outlying Scores
      On The Mean
      On The Median
      On The Mode
   How To Calculate Central Tendency
      Calculating The Mean
      Calculating The Median
      When There’s No Middle-Most Score
      Calculating The Mode
   Formulas For Central Tendency
   Basic Facts About Central Tendency
   Vocabulary
   Quiz 2
   Summary

Mode & Outlying Scores

November 5, 2008 by  
Filed under Even More

The mode of the following distribution  is 5:

5
5
3
4
5
6  
7

Note, that the mode is not impacted by outlying scores. The mode of the following also is 5:

700
    5
    5
    3
    4
    5
    6
    7

NOW YOU CHOOSE:
   Day 2: Central Tendency
   
A Bit More About Central Tendency
   Even More About Central Tendency
   More Examples
      More Mean Examples

      More Median Examples
      Median Is Middle Of Distribution
      More Mode Examples
   Impact of Outlying Scores
      On The Mean
      On The Median
      On The Mode
   How To Calculate Central Tendency
      Calculating The Mean
      Calculating The Median
      When There’s No Middle-Most Score
      Calculating The Mode
   Formulas For Central Tendency
   Basic Facts About Central Tendency
   Vocabulary
   Quiz 2
   Summary

 

 

 

More Median

November 5, 2008 by  
Filed under Even More

The median of the following numbers is 12:

22
14
12
  4 
  3

The median of the following is 9:

  6
14
  3
44 
  9

(Remember to order the numbers from highest to lowest before finding the middle score)

The median of the following is 7:

24
17
11 
  9
  5 
  4 
  3 
  2

NOW YOU CHOOSE:
   Day 2: Central Tendency
   
A Bit More About Central Tendency
   Even More About Central Tendency
   More Examples
      More Mean Examples

      More Median Examples
      Median Is Middle Of Distribution
      More Mode Examples
   Impact of Outlying Scores
      On The Mean
      On The Median
      On The Mode
   How To Calculate Central Tendency
      Calculating The Mean
      Calculating The Median
      When There’s No Middle-Most Score
      Calculating The Mode
   Formulas For Central Tendency
   Basic Facts About Central Tendency
   Vocabulary
   Quiz 2
   Summary

 

Middle Of Distribution

November 5, 2008 by  
Filed under Even More

The median is the middle of the distribution, not the middle of the raw scores.

The median of the following numbers is 7, not 11:

14
12
  4
11
  3
  7
  5

First the scores must be put in order, then the middle-most score found. So, the previous scores would make the following distribution:

14
12
11
  7
  5
  4
  3

And the median is 7.

NOW YOU CHOOSE:
   Day 2: Central Tendency
   
A Bit More About Central Tendency
   Even More About Central Tendency
   More Examples
      More Mean Examples

      More Median Examples
      Median Is Middle Of Distribution
      More Mode Examples
   Impact of Outlying Scores
      On The Mean
      On The Median
      On The Mode
   How To Calculate Central Tendency
      Calculating The Mean
      Calculating The Median
      When There’s No Middle-Most Score
      Calculating The Mode
   Formulas For Central Tendency
   Basic Facts About Central Tendency
   Vocabulary
   Quiz 2
   Summary

 

 

 

Median & Outlying Scores

November 5, 2008 by  
Filed under Even More

The median is not impacted by outlying scores. It is affected by adding or subtracting a score but not from changing an end score to a larger number. Notice that the median goes unchanged when the surrounding scores are changed.

The median for these scores is 5:

7
6
5
5
5
4
3

The median of these scores also is 5:

700 
   6 
   5 
   5 
   5 
   4 
   3

The median of the following data is also 7:

   164 
   102 
      7 
      1 
      1

Remember, unlike means, medians are not affected by outlying scores. The median is simply the middle-most point, regardless of who surrounds it.

NOW YOU CHOOSE:
   Day 2: Central Tendency
   
A Bit More About Central Tendency
   Even More About Central Tendency
   More Examples
     
More Mean Examples

      More Median Examples
      Median Is Middle Of Distribution
      More Mode Examples
   Impact of Outlying Scores
      On The Mean
      On The Median
      On The Mode
   How To Calculate Central Tendency
      Calculating The Mean
      Calculating The Median
      When There’s No Middle-Most Score
      Calculating The Mode
   Formulas For Central Tendency
   Basic Facts About Central Tendency
   Vocabulary
   Quiz 2
   Summary

 

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