Study Notes on Descriptive Statistics, Hypothesis Testing, and Normality

Commonly Reported Descriptive Statistics

Descriptive statistics provide a summary of data gathered from a sample, focusing on central tendency and dispersion. The most common metrics include:

• Mean: Represents the average or the usual score of respondents on a studied variable. It is a descriptive statistic that is sensitive to every value in the distribution.
• Standard Deviation (SD): Refers to the usual dispersion or spread of scores in reference to the mean.
• Verbal Interpretation (VI): A category used to tag yielding average values. This is accomplished using a transmutation table or "Cut-Off Scores" to give qualitative meaning to quantitative data.

Writing Descriptive Results: Case Studies

Sample Experiment 1: Logic Quizzes and Competition Types

A team of experimental psychologists sought to identify whether group or individual competition catalyzes better performance in logic quizzes. The study focused on 3rd-year philosophy majors at a college in Pangasinan. The administration provided $20$ students from each of the four sections, totaling $N=80$.

Data Summary:

• Group Competition: $N=40$, Mean Logic Score = $23.45$, $SD=3.2$, Verbal Interpretation = "Passed".
• Individual Competition: $N=40$, Mean Logic Score = $37.02$, $SD=2.9$, Verbal Interpretation = "High Passed".

Written Presentation Example: Based on the results of the descriptive statistics implemented, the philosophy students subjected to individual competition yielded numerically higher scores in the logic quiz. They achieved a mean score of $37.02$ ($SD=2.9$), interpreted as "High Passed". Conversely, students in group competition garnered an average score of $23.45$ ($SD=3.2$), which is interpreted as "Passed".

Sample Study 2: Selfie Posting and Heterosexuality

A personality psychologist investigated if the frequency of selfies posted per month on Social Networking Sites (FB, Twitter, Instagram) by male young adults relates to their level of heterosexuality. A survey of $356$ male university students in the U-belt area was conducted.

Data Summary ($N=356$):

• Selfie Posting: Mean = $2.41$, $SD=2.67$, Verbal Interpretation = "Low".
• Heterosexuality: Mean = $3.97$, $SD=0.36$, Verbal Interpretation = "High".

Written Presentation Example: Based on the descriptive statistics for the $356$ male university students, the average frequency of selfie posting per month is $2.41$ ($SD=2.67$), interpreted as "Low". This indicates that selfie posting among male U-Belt students is not a common practice. In terms of heterosexuality, the mean level is $3.97$ ($SD=0.36$), which is considered "High". This indicates that the respondents are commonly males with sexual desires for the opposite gender.

Research Questions and the Statement of the Problem

Research questions serve as a compass for data gathering and statistical treatment. They are the specific list of questions a study must answer based on the sample data.

Formulating Research Questions

1. Identify the Concept: Choose a general concept and a theory (e.g., Beck's Depression Model) that explains relationships between variables. Look for literature "blindspots".
2. Create a Specific Title: Example: "Rumination and its relationship on Depression".
3. Structure the Statement of the Problem (SOP): The SOP must include:
   • Descriptive Statistics questions.
   • Inferential Statistics questions.
   • Implication questions.

Example SOP for Rumination and Depression

• General Problem: To evaluate the relationship of Rumination to Depression.
• Specific Questions:
  • 1. What is the level of the participants' Rumination?
  • 2. What is the level of the participants' Depression?
  • 3. Is Rumination related to the depression experienced by the participants?
  • 4. What implications can be drawn from the results of the study?

Hypothesis Making and Testing

A hypothesis is an educated guess for questions requiring inferential statistics.

Types of Hypotheses

• Null Hypothesis ($H_0$): The logical counterpart to the alternative hypothesis. It specifies there is no effect or that an effect exists in the direction opposite to the alternative.
• Alternative Hypothesis ($H_1$): Claims that the differences in results are due to the independent variable.
  • Directional Alternative: Specifies the type of effect (e.g., "Rumination is positively related to Depression").
  • Non-Directional Alternative: Does not specify the direction of the effect (e.g., "Rumination is related to Depression").

P-Value Method for Hypothesis Testing

1. Formulate the hypothesis and identify the claim.
2. Determine the Critical Value.
3. Compute the Test value using appropriate statistics.
4. Make a decision (Reject or Fail to Reject $H_0$).
5. Summarize the results.

Interpretive Decision Rules

• If $p < \alpha$: Reject $H_0$.
• If $p > \alpha$: Fail to reject $H_0$.

Errors in Data Interpretation

• Type I Error: The decision to reject the null hypothesis when it is actually true (claiming an effect exists when it does not).
• Type II Error: The decision to retain (fail to reject) the null hypothesis when it is actually false (claiming no effect exists when it does).

Inferential Statistics: Parametric vs. Non-Parametric

Inferential statistics allow researchers to reach conclusions extending beyond immediate data, determining if scores are significant or occurred by chance.

Common Statistical Treatments and Purposes

• Independent T-Test: Comparing two groups to identify the effect of the Independent Variable (IV) on the Dependent Variable (DV).
• Dependent T-Test: Comparing two situations for a single group to identify the effect of an IV on a DV.
• One-Way ANOVA: Comparing three or more groups.
• Two-Way ANOVA: Identifying effects of two or more IVs on a DV through multiple comparisons.
• Pearson Correlation: Identifying relationships between two or more interval/ratio variables.
• Spearman’s Rho (Non-Parametric): Identifying relationships between two or more ordinal variables using deviations.
• Kendall’s Tau (Non-Parametric): Identifying relationships between two or more ordinal variables using concordant and discordant pairs.
• Linear Regression: Predicting a DV based on a specific IV.
• Multiple Regression: Predicting a DV based on a specific model/multiple IVs.
• Chi-Square (Non-Parametric): Identifying relationships between nominal or categorical variables.

The Normal Curve (Gaussian Distribution)

The Normal Curve is a theoretical bell-shaped probability distribution of population scores described by a specific equation:

$Y = \frac{N}{\sqrt{2\pi}\sigma} e^{-(x-\mu)^2 / 2\sigma^2}$

• $Y$: Frequency of a given value of $X$
• $X$: Any score in the distribution
• $\mu$: Mean of the distribution
• $\sigma$: Standard deviation
• $N$: Total frequency
• $\pi$: Constant ($3.1416$)
• $e$: Constant ($2.7183$)

Important Characteristics

1. The Mean, Median, and Mode are all equal.
2. The distribution is symmetrical on both sides of the median.
3. The tails are asymptotic, meaning they approach but never meet the horizontal axis.
4. Variables in behavioral science closely resemble the normal curve, and it is a requirement for many inferential tests.

The Empirical Rule (68-95-99.7 Rule)

This rule determines the proportion of values within specific distances from the mean:

• $68.27\%$ of values fall within $\pm 1$ SD of the mean.
• $95.45\%$ of values fall within $\pm 2$ SD of the mean.
• $99.73\%$ of values fall within $\pm 3$ SD of the mean.

Normality Testing with SPSS and JAMOVI

Data should be tested against the normal curve using the Kolmogorov-Smirnov or Shapiro-Wilk tests. To establish normality, the Significance ($p$) value should be greater than $0.05$.

Performing Normality Tests in SPSS

1. Go to Analyze > Descriptive Stats > Explore.
2. Place the data array into the Dependent List.
3. Select Plots and check Normality Plots with Test.
4. Click OK.
5. Interpretation: If Sig. is $> .05$, the distribution resembles a Normal Curve. If Sig. is $< .05$, the data is significantly different from a normal curve (Not Normal).

Producing a Histogram with Normal Curve in SPSS

1. Go to Analyze > Descriptive Stats > Frequencies.
2. Select Charts.
3. Choose Histogram and check Show Normal Curve on Histogram.

Performing Normality Tests in JAMOVI

1. Go to Analyses > Exploration > Descriptives.
2. Add the target variable to the Variables box.
3. Under the Statistics dropdown, check Shapiro-Wilk.
4. The output will display the Shapiro-Wilk W and Shapiro-Wilk p results.

Practical Problems and Significance Examples

• Weight Study in Baliuag: A nutritionist believes the true mean weight of residents is different from $168\,lbs$. In a sample of $36$ individuals, the mean was $169.5\,lbs$ ($SD=3.9$). At a $95\%$ confidence level, a test is performed to determine if the null hypothesis can be discarded.
• Machine Fluid Dispensing: A machine dispenses $80\,ml$. An employee measures a sample of $40$ bottles, finding a mean of $78\,ml$ ($SD=2.5$). Test at a $95\%$ confidence level to see if the machine is failing.
• Battery Lifespan: Average lifespan is $\ge 2$ years. An engineer tests $10$ samples and finds the mean is $1.8$ years ($SD=0.15$). Test at a $99\%$ confidence level to see if there is enough evidence to discard the null hypothesis.