3: Statistical concepts and test score interpretation

Question 1

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What are the 3 important Properties of Scales?

Answer 1

Properties of Scales
Three important properties make scales of measurement different from one another: magnitude, equal intervals, and an absolute 0.

Question 2

Describe ‘Magnitude’?

Answer 2

Magnitude
Magnitude is the property of “moreness.” A scale has the property of magnitude if we can say that a particular instance of the attribute represents more, less, or equal amounts of the given quantity than does another instance (Gravetter & Wallnau, 2016; Howell, 2008; McCall, 2001). On a scale of height, for example, if we can say that John is taller than Fred, then the scale has the property of magnitude. A scale that does not have this property arises, for example, when a gym coach assigns identification numbers to teams in a league (team 1, team 2, etc.). Because the numbers only label the teams, they do not have the property of magnitude. If the coach were to rank the teams by the number of games they have won, then the new numbering system (games won) would have the property of magnitude.

Question 3

Describe ‘Equal Intervals’?

Answer 3

Equal Intervals
The concept of equal intervals is a little more complex than that of magnitude. A scale has the property of equal intervals if the difference between two points at any place on the scale has the same meaning as the difference between two other points that differ by the same number of scale units. For example, the difference between inch 2 and inch 4 on a ruler represents the same quantity as the difference between inch 10 and inch 12: exactly 2 inches.

As simple as this concept seems, a psychological test rarely has the property of equal intervals. For example, the difference between intelligence quotients (IQs) of 45 and 50 does not mean the same thing as the difference between IQs of 105 and 110. Although each of these differences is 5 points

(50-45=5 and 110-105=5)

the 5 points at the first level do not mean the same thing as 5 points at the second. We know that IQ predicts classroom performance. However, the difference in classroom performance associated with differences between IQ scores of 45 and 50 is not the same as the differences in classroom performance associated with IQ score differences of 105 and 110. In later chapters, we will discuss this problem in more detail.

When a scale has the property of equal intervals, the relationship between the measured units and some outcome can be described by a straight line or a linear equation in the form

𝑌 = 𝑎 + 𝑏𝑋.

This equation shows that an increase in equal units on a given scale reflects equal increases in the meaningful correlates of units. For example, Figure 2.1 shows the hypothetical relationship between scores on a test of manual dexterity and ratings of artwork. Notice that the relationship is not a straight line. By examining the points on the figure, you can see that at first the relationship is nearly linear: Increases in manual dexterity are associated with increases in ratings of artwork. Then the relationship becomes nonlinear. The figure shows that after a manual dexterity score of approximately 5, increases in dexterity produce relatively small increases in quality of artwork.

Question 4

Describe ‘Absolute 0’?

Answer 4

Absolute 0

An absolute 0 is obtained when nothing of the property being measured exists. For example, if you are measuring heart rate and observe that your patient has a rate of 0 and has died, then you would conclude that there is no heart rate at all.

For many psychological qualities, it is extremely difficult, if not impossible, to define an absolute 0 point. For example, if one measures shyness on a scale from 0 through 10, then it is hard to define what it means for a person to have absolutely no shyness (McCall, 2001).

Question 5

Define four scales of measurement…

Answer 5

Nominal, Ordinal, Interval and Ratio

Question 6

Describe a Nominal scale?

Answer 6

You can see that a nominal scale does not have the property of magnitude, equal intervals, or an absolute 0. Nominal scales are really not scales at all; their only purpose is to name objects. For example, the numbers on the backs of football players’ uniforms are nominal. Nominal scales are used when the information is qualitative rather than quantitative. Social science researchers commonly label groups in sample surveys with numbers

(such as 1=African American, 2=white, and 3=Mexican American).

When these numbers have been attached to categories, most statistical procedures are not meaningful. On the scale for ethnic groups, for instance, what would a mean of 1.87 signify?

Question 7

Describe ‘Ordinal’ scale?

Answer 7

A scale with the property of magnitude but not equal intervals or an absolute 0 is an ordinal scale. This scale allows you to rank individuals or objects but not to say anything about the meaning of the differences between the ranks. If you were to rank the members of your class by height, then you would have an ordinal scale. For example, if Fred was the tallest, Susan the second tallest, and George the third tallest, you would assign them the ranks 1, 2, and 3, respectively. You would not give any consideration to the fact that Fred is 8 inches taller than Susan, but Susan is only 2 inches taller than George.

For most problems in psychology, the precision to measure the exact differences between intervals does not exist. So, most often one must use ordinal scales of measurement. For example, IQ tests do not have the property of equal intervals or an absolute 0, but they do have the property of magnitude. If they had the property of equal intervals, then the difference between an IQ of 70 and one of 90 should have the same meaning as the difference between an IQ of 125 and one of 145. Because it does not, the scale can only be considered ordinal. Furthermore, there is no point on the scale that represents no intelligence at all—that is, the scale does not have an absolute 0.

Question 8

Describe ‘interval’ scale?

Answer 8

When a scale has the properties of magnitude and equal intervals but not absolute 0, we refer to it as an interval scale. The most common example of an interval scale is the measurement of temperature in degrees Fahrenheit. This temperature scale clearly has the property of magnitude, because
35F° is warmer than 32F°, 65F° is warmer than 64F°, and so on. Also, the difference between 90F° and 80F° is equal to a similar difference of 32°
at any point on the scale. However, on the Fahrenheit scale, temperature does not have the property of absolute 0. If it did, then the 0 point would be more meaningful. As it is, 0 on the Fahrenheit scale does not have a particular meaning. Water freezes at

32F° and boils at 212F°.

Because the scale does not have an absolute 0, we cannot make statements in terms of ratios. A temperature of

22F° is not twice as hot as 11F°, and 70F° is not twice as hot as 35F°.

The Celsius scale of temperature is also an interval rather than a ratio scale. Although 0 represents freezing on the Celsius scale, it is not an absolute 0. Remember that an absolute 0 is a point at which nothing of the property being measured exists. Even on the Celsius scale of temperature, there is still plenty of room on the thermometer below 0. When the temperature goes below freezing, some aspect of heat is still being measured.

Question 9

Describe ‘Ratio’ scale?

Answer 9

A scale that has all three properties (magnitude, equal intervals, and an absolute 0) is called a ratio scale. To continue our example, a ratio scale of temperature would have the properties of the Fahrenheit and Celsius scales but also include a meaningful 0 point. There is a point at which all molecular activity ceases, a point of absolute 0 on a temperature scale. Because the Kelvin scale is based on the absolute 0 point, it is a ratio scale:

22K° is twice as cold as 44K°.

Examples of ratio scales also appear in the numbers we see on a regular basis. For example, consider the number of yards gained by running backs on football teams. Zero yards actually means that the player has gained no yards at all. If one player has gained 1000 yards and another has gained only 500, then we can say that the first athlete has gained twice as many yards as the second.

Another example is the speed of travel. For instance, 0 miles per hour (mph) is the point at which there is no speed at all. If you are driving onto a highway at 30 mph and increase your speed to 60 when you merge, then you have doubled your speed.

Question 10

Describe Level of measurement in relation to nominal data and give an example…

Answer 10

Level of measurement is important because it defines which mathematical operations we can apply to numerical data (Streiner, Norman, & Cairney, 2014).

For nominal data, each observation can be placed in only one mutually exclusive category.

For example, you are a member of only one gender. One can use nominal data to create frequency distributions (see the next section), but no mathematical manipulations of the data are permissible.

Question 11

Describe level of measurements in relation to ordinal measerements and give example…

Answer 11

Ordinal measurements can be manipulated using arithmetic; however, the result is often difficult to interpret because it reflects neither the magnitudes of the manipulated observations nor the true amounts of the property that have been measured.

For example, if the heights of 15 children are rank ordered, knowing a given child’s rank does not reveal how tall he or she stands. Averages of these ranks are equally uninformative about height.

Question 12

Describe level of measurements in relation to interval data and give example…

Answer 12

With interval data, one can apply any arithmetic operation to the differences between scores. The results can be interpreted in relation to the magnitudes of the underlying property. However, interval data cannot be used to make statements about ratios.

For example, if IQ is measured on an interval scale, one cannot say that an IQ of 160 is twice as high as an IQ of 80. This mathematical operation is reserved for ratio scales, for which any mathematical operation is permissible (Gravetter & Wallnau, 2016).

Question 13

Describe ‘frequency distribution’?

Answer 13

frequency distribution
The systematic arrangement of scores on a variable or a measure to reflect how frequently each value occurred.

Question 14

Describe the distribution of scores…

Answer 14

A single test score means more if one relates it to other test scores. A distribution of scores summarizes the scores for a group of individuals. In testing, there are many ways to record a distribution of scores.

The frequency distribution displays scores on a variable or a measure to reflect how frequently each value was obtained. With a frequency distribution, one defines all the possible scores and determines how many people obtained each of those scores.

Usually, scores are arranged on the horizontal axis from the lowest to the highest value. The vertical axis reflects how many times each of the values on the horizontal axis was observed.

For most distributions of test scores, the frequency distribution is bell shaped, with the greatest frequency of scores toward the center of the distribution and decreasing scores as the values become greater or less than the value in the center of the distribution.

Question 15

Describe the graph (Figure 2.2) and what it means

Answer 15

Figure 2.2 shows a frequency distribution of 1000 observations that takes on values between 61 and 90.

Notice that the most frequent observations fall toward the center of the distribution, around 75 and 76. As you look toward the extremes of the distribution, you will find a systematic decline in the frequency with which the scores occur.

For example, the score of 71 is observed less frequently than 72, which is observed less frequently than 73, and so on. Similarly, 78 is observed more frequently than 79, which is noted more often than 80, and so forth.

Though this neat symmetric relationship does not characterize all sets of scores, it occurs frequently enough in practice for us to devote special attention to it. We explain this concept in greater detail in the section on the normal distribution.

Question 16

Look at figures 2.2 - 2.5 describe and explain…

Answer 16

Table 2.2 lists the rainfall amounts in San Diego, California, between 1965 and 2015.

Figure 2.3 is a histogram based on the observations. The distribution is slightly skewed, or asymmetrical.

We say that Figure 2.3 has a positive skew because the tail goes off toward the higher or positive side of the X axis.

There is a slight skew in Figures 2.3 and 2.4, but the asymmetry in these figures is relatively hard to detect.

Figure 2.5 gives an example of a distribution that is clearly skewed. The figure summarizes annual household income in the United States at the time of the last census in 2010.

Very few people make high incomes, while the great bulk of the population is bunched toward the low end of the income distribution. Of particular interest is that this figure only includes household incomes less than $100,000.

For household incomes greater than $100,000, the government only reports incomes using class intervals of $50,000. In 2011, about 16% of the U.S. households had incomes greater than $100,000.

Because some households have extremely high incomes, you can imagine that the tail of this distribution would go very far to the right. Thus, income is an example of a variable that has positive skew.

Question 17

Describe frequency polygon (see Figure 2.4)

Answer 17

One can also present this same set of data as a frequency polygon (see Figure 2.4). Here the amount of rainfall is placed on the graph as a point that represents the frequencies with which each interval occurs. Lines are then drawn to connect these points.

Whenever you draw a frequency distribution or a frequency polygon, you must decide on the width of the class interval. The class interval for inches of rainfall is the unit on the horizontal axis. For example, in Figures 2.3 and 2.4, the class interval is 3 inches—that is, the demarcations along the X axis increase in 3-inch intervals. This interval is used here for convenience; the choice of 3 inches is otherwise arbitrary.

Question 18

Describe Percentile ranks…

Answer 18

Percentile Ranks

Percentile ranks replace simple ranks when we want to adjust for the number of scores in a group.

A percentile rank answers the question, “What percent of the scores fall below a particular score
(𝑋𝑖)?” To calculate a percentile rank, you need only follow these simple steps:

(1)
determine how many cases fall below the score of interest,

(2)
determine how many cases are in the group,

(3)
divide the number of cases below the score of interest (Step 1) by the total number of cases in the group (Step 2), and

(4)
multiply the result of Step 3 by 100.

The formula is

𝑃𝑟 = 𝐵/N × 100 = percentile rank of 𝑋𝑖

where

𝑃𝑟 = percentile rank
𝑋𝑖 = the score of interest
𝐵 = the number of scores below 𝑋𝑖
N = the total number of scores

This means that you form a ratio of the number of cases below the score of interest and the total number of scores. Because there will always be either the same or fewer cases in the numerator (top half) of the equation than there are in the denominator (bottom half), this ratio will always be less than or equal to 1. To get rid of the decimal points, you multiply by 100.

Question 19

Give an example of Percentile ranks…

Answer 19

As an example, consider the runner who finishes 62nd out of 63 racers in a gym class. To obtain the percentile rank, divide 1 (the number of people who finish behind the person of interest) by 63 (the number of scores in the group). This gives you 1/63, or .016. Then multiply this result by 100 to obtain the percentile rank, which is 1.6. This rank tells you the runner is below the 2nd percentile.

Question 20

In regards to the previous question calculate the percentile rank…

Answer 20

Now consider the Bay to Breakers race, which attracts 50,000 runners to San Francisco. If you had finished 62nd out of 50,000, then the number of people who were behind you would be 49,938. Dividing this by the number of entrants gives you .9988. When you multiply by 100, you get a percentile rank of 99.88. This tells you that finishing 62nd in the Bay to Breakers race is exceptionally good because it places you in the 99.88th percentile.

Answer 21

Psychological Testing in Everyday Life 2.1 presents the calculation of percentile ranks of the infant mortality rates of selected countries as reported by the Central Intelligence Agency (CIA). Infant mortality is defined as the number of babies out of 1000 who are born alive but die before their first birthday. Before proceeding, we should point out that the meaning of this calculation depends on which countries are used in the comparison.