Variables, Z Scores, Population, And Output

Variables, Z Scores, Population and Output

Herve Ngate.quantitative Research Methods Busi 820 B05may 23 2024disc

Herve Ngate Quantitative Research Methods, BUSI 820-B05 May 23, 2024 Discussion Week 2: Variables, Z Scores, Population and Output D2.3.1 “If you have categorical, ordered data (such as low income, middle income, high income) what type of measurement would you have? Why?†According to Morgan et al. (2020), if you have categorical, ordered data such as low income, middle income, and high income, you will have an ordinal level of measurement because ordinal data is a type of categorical data with a natural order or ranking. In this case, the income categories have a clear order from low to middle income and then to high income. This type of measurement allows for a meaningful comparison of the income categories based on their relative positions in the order.

D2.3.2 (a) Compare and contrast nominal, dichotomous, ordinal, and normal variables. - Nominal variables are categorical variables with no inherent order or ranking. Examples include gender, ethnicity, or marital status. - Dichotomous variables are a specific type of nominal variable with only two categories. Examples include yes/no responses, true/false answers, or presence/absence of a characteristic (Shreffler and Huecker, 2023). - Ordinal variables are categorical variables with a natural order or ranking. Examples include Likert scale responses (e.g., strongly disagree, disagree, neutral, agree, strongly agree) or educational levels (e.g., high school, college, graduate school). - Normal variables are measured on a scale with equal intervals between values and have a meaningful zero point. These variables can take any numerical value and are typically used for quantitative measurements such as height, weight, or temperature.

D2.3.2 (b) In social science research, why isn't it important to distinguish between interval and ratio variables? In social science research, it is not always essential to distinguish between interval and ratio variables because they involve quantitative measurement. In addition, both variables can often be treated similarly in statistical analysis when using them with the term scale in SPSS (Morgan et al., 2020). Both interval and ratio variables can be subjected to statistical techniques such as mean, standard deviation, correlation, and regression analysis.

D2.3.3 What percent of the area under the standard normal curve is within one standard deviation of (above or below) the mean? What does this tell you about scores that are more than one standard deviation away from the mean? The standard normal distribution follows a bell-shaped "normal curve". In this distribution, based on the Empirical Rule around 68% of the zone “under the curve falls within one standard deviation of the mean†(Ross, 2021). Therefore, considering the area under the standard normal curve, approximately 34% is within one standard deviation above the mean, and approximately 34% is within one standard deviation below the mean. These two percentages sum up to a total of roughly 68%. Scores more than one standard deviation away from the mean are relatively rare, falling into the tails of the distribution, indicating they are significantly different from the average.

D2.3.4 (a) How do z scores relate to the normal curve? The normal curve, also known as the standard normal distribution or Z-distribution, is a specific type of distribution with a mean of 0 and a standard deviation of 1. The z-scores correspond to specific percentiles on the normal curve, enabling us to determine the proportion of scores below or above a particular z-score. Therefore, Z-scores relate to the normal curve by indicating the number of standard deviations a specific score is from the mean.

D2.3.4 (b) How would you interpret a z score of -3.0? A z-score of -3.0 indicates that the raw score is three standard deviations below the mean. In other words, it suggests that the score is significantly lower than the mean value. Since the normal distribution is symmetrical, a z-score of -3.0 indicates that the raw score is in the left tail of the distribution, far away from the mean.

D2.3.4 (c) What percentage of scores is between a z of -2 and a z of +2? The Empirical Rule states that approximately “95% of the scores fall within two standard deviations of the mean in a normal distribution†(Ross, 2021). Therefore, between a z-score of -2 and a z-score of +2, we can expect approximately 95% of the scores to lie within this range. This means 95% of the scores are within two standard deviations above and below the mean. This highlights the concentration of data points around the average in a normal distribution.

D2.3.5 “Why should you not use a frequency polygon if you have nominal data? What would be better to use to display nominal data?†A frequency polygon represents the connection of midpoints of each category's bar in a histogram. According to Morgan et al. (2020), you should not use a frequency polygon to display nominal data because nominal data represents categories or groups with no inherent order or ranking. Connecting the midpoints could create a misleading visual representation. A better option is to use a bar chart to display nominal data (Morgan et al., 2020). This graphical representation uses rectangular bars to represent the frequencies or proportions of each category, allowing for easy comparison between categories.

Paper For Above instruction

Understanding variables and their measurement is fundamental in quantitative research, especially when utilizing statistical tools like SPSS. Correct identification of variable types ensures data is appropriately analyzed, leading to valid conclusions. In social sciences, where data types vary from categorical to continuous, specificity in measurement levels influences the choice of statistical tests, visualization tools, and interpretation.

When dealing with ordered categories such as income levels—low, middle, and high—the appropriate measurement scale is ordinal. Morgan et al. (2020) articulate that ordinal variables possess a natural ranking but lack equal intervals between categories. This ordinal scale allows researchers to analyze the relative positioning of income categories without implying precise quantitative differences. Such categorization facilitates meaningful comparisons, such as determining whether higher income categories are associated with specific outcomes or behaviors. The importance of ordinal measurement lies in its ability to preserve the inherent order in the data without assuming uniform spacing, thus maintaining the integrity of the analysis.

In contrasting nominal, dichotomous, ordinal, and normal variables, it is crucial to recognize their unique properties. Nominal variables refer to categories with no intrinsic order, exemplified by gender or ethnicity, captured via labels or numbers for coding purposes. Dichotomous variables are a subset of nominal variables with precisely two categories, such as yes/no responses or presence/absence indicators. Ordinal variables, as previously described, have a meaningful order but uneven intervals, such as Likert scale responses. Normal variables typically refer to continuous data measured on interval or ratio scales, which incorporate equal spacing and a true zero point, like height or temperature. Recognizing these distinctions informs the selection of appropriate analytical methods and graphical representations.

The differentiation between interval and ratio variables is often less critical in social science research because both involve numerical data suitable for analysis with similar statistical techniques. Morgan et al. (2020) explain that interval variables feature equal units between measurements but lack a true zero, making ratios meaningless (e.g., temperature in Celsius). Ratio variables, however, have a meaningful zero point and allow for ratio-based calculations like multiplication or division, as seen in income or weight. While technically distinct, both types are often treated similarly during analysis—computing means, standard deviations, or correlations—especially when the data is roughly normally distributed. Consequently, the practical impact of distinguishing between these two measurement levels is limited in many social science contexts.

Regarding the properties of the normal distribution, approximately 68% of scores reside within one standard deviation of the mean, as established by the Empirical Rule. This concentration illustrates the central tendency's dominance in data distribution, with the remaining scores spread across the tails. Scores beyond one standard deviation are rarer and indicate deviations from typical behavior or characteristics.

In relation to z-scores, which standardize individual scores relative to the mean and standard deviation, each z-value corresponds to a specific position on the normal curve. A z-score of -3.0 indicates that the raw score is three standard deviations below the mean, an event that is quite uncommon and typically considered an outlier. Interpreting z-scores involves understanding their relation to percentile ranks and the likelihood of occurrence within the distribution.

The percentage of scores falling between z-scores of -2 and +2 encompasses approximately 95% of the data in a normal distribution, highlighting the concentration around the mean. Because of this, scores outside this range—beyond ±2 standard deviations—are less frequent, emphasizing their significance as potential outliers or extreme cases.

Selection of appropriate graphical tools hinges on the nature of the data. Frequency polygons, suitable for continuous or interval data, connect midpoints of class intervals to display trends. However, for nominal data—categories without a natural order—frequency polygons are ineffective as they imply a continuum where none exists. Instead, bar graphs are preferred, as they represent discrete categories with rectangular bars, facilitating straightforward comparisons of category frequencies or proportions without suggesting a numerical or ordered relationship.

In synthesis, accurately understanding and identifying data types and measurement levels underpin rigorous quantitative analysis. It ensures the utilization of correct statistical methods and visualization techniques, enabling researchers to derive valid insights from their data. Proper alignment between data characteristics and analytical strategies enhances the clarity and credibility of research findings in the social sciences.

References

• Morgan, G., Leech, N., Gloeckner, G., & Barrett, K. (2020). IBM SPSS for Introductory Statistics: Use and interpretation (6th ed.). Routledge.
• Ross, S. M. (2021). Descriptive statistics. Elsevier.
• Shreffler, J., & Huecker, M. R. (2023). Types of variables and commonly used statistical designs. StatPearls - NCBI Bookshelf.
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