**D2.1.a Compare and Contrast**

There are four levels of measurement with respect to variables which are; nominal, dichotomous, ordinal, and normal (Morgan, Leech, Gloeckner, & Barrett, 2013). Nominal variables have characteristics of having three or more levels of measurement, not ordered, and consist of names or labels (Morgan et al., 2013). An example of a nominal variable would be “religion” whereby a 1 could represent catholic, a 2 could represent protestant, and a 3 could represent other (Morgan et al., 2013). Dichotomous variables have characteristics of having two levels of measurement that can be either ordered or unordered (Morgan et al., 2013). An example of a dichotomous variable would be “science scores (high versus low)” (Morgan et al., 2013). Ordinal values have characteristics of having three or more levels of measurement that are ordered and have unequal intervals between levels that are not normally distributed and are often skewed (Morgan et al., 2013). An example of ordinal variables would be “Father’s education” (Morgan et al., 2013). Normal variables have characteristics of having five or more levels of measurement that are ordered and that are approximately normally distributed with equal intervals between levels (Morgan et al., 2013). An example of a normal variable could be weight (Morgan et al., 2013).

**D2.1.b Interval and Ratio Data**

Interval variables are defined as having ordered categories that have equal spacing between them such as the physical measurement of “length” (Morgan et al., 2013). Length can be measured so that there can be 1 inch, 2 inches, or 3inches (i.e. there exists an interval between the measurements) (Morgan et al., 2013). Length also can have a zero length which illustrates that a “length” measurement can be a ratio variable because a true absolute zero value exists (Morgan et al., 2013). In the social science arena, a variable such as extroversion can be measured but there does not exist an absolute zero value for degree of extroversion so the distinction between ratio and interval is not important in social science research (Morgan et al., 2013).

**D2.2 Z Scores and the Population**

**D2.2.a Relate to Normal Curve**

A normal curve is unimodal having one “hump” in the middle of the curve which is represented by the most frequent value (Morgan et al., 2013). One standard deviation above or below the mean is represented by approximately 34% of the area under the normal curve with 68% of the area under the normal curve being within one standard deviation from the mean (Morgan et al., 2013). Z scores are given underneath the normal curve whereby one standard deviation is referred to as a z score of 1 in the positive direction or a z score of -1 in the negative direction (Morgan et al., 2013). Similarly, two standard deviations from the mean represents an area under the normal curve of 47.5% (z score = 2) (Morgan, et al., 2013).

**D2.2.b Interpretation of a z Score of -3.0**

A z score of -3 would be interpreted that a data point is three standard deviations or 49.5% below the mean value of the normal curve (Morgan et al., 2013).

**D2.2.c Percentage of Scores Between z of -2 and z of +2**

95% of scores are between a z of -2 and a z of +2 (Morgan et al., 2013). This is important because if one subtracts the 95% from 100% the remainder of 5% relates to the p value of 0.05 which is needed for statistical significance (Morgan et al., 2013). Values that are outside of this 95% range are deemed to be rare events (Morgan et al., 2013).

**D2.3 Interpreting Output 4.1b**

In the output 4.1b Descriptive Statistics table, “skewness statistic” is given in the right-most section of the table corresponding to each of the variables given on the left-most side of the table (Morgan et al., 2013). The only skewness statistic that is more than 1.00 or less than -1.00 is the value for “Competence scale” which is -1.634. This answer is important because if the skewness is more than +1.0 or less than -1.0 the distribution is skewed with a negative value indicating a long tail to the left. Skewness is an index that aids in determining the degree to which a variable’s distribution deviates from that of the normal curve (Morgan et al., 2013). This agrees with the boxplot given in Output 4.2 because there are four low outliers given in the competence scale which indicates that the distribution is skewed with a tail to the left indicating a negative skew value (Morgan et al., 2013).

**D2.4 Interpreting Output 4.4**

**D2.4.a Interpret the Means**

The variables in Output 4.4 are all dichotomous variables which means there are two categories for each variable and an interpretation of the mean can be made for each variable. By knowing what “0” and “1” represents from the “Variables View” tab or the codebook an interpretation of the mean can be made. The interpretations are; 55% female, 79% took algebra 1 in high school, 47% took algebra 2 in high school, 48% took geometry in high school, 27% took trigonometry in high school, 11% took calculus in high school, and 41% had math grades that were mostly A-B (Morgan et al., 2013).

**D2.4.b Total Participants**

There are 75 total participants (Morgan et al., 2013).

**D2.4.c Complete Data**

There are 75 participants that have complete data (i.e. no missing data) (Morgan et al., 2013).

**D2.4.d Percentage Male**

There are 45% of the participants that are male (Morgan et al., 2013).

**D2.4.e Percentage Took Algebra**

There were 79% that took Algebra 1 in High School and 47% that took Algebra 2 in High School (Morgan et al., 2013).