NSE212W7 Analyzing Quantitative Data-Descriptive & Bivariate Statistics, 2024

Descriptive Statistics Singh & Thirsk (2022), Chapter 17, pp. 368-69; pp. 372-377 Salkind & Frey, 2020; Chapter 2, p.19-36; Chapter 3, p.41-53; Chapter 4, p.56-74 Adapted from Salkind, Statistics for People Who (Think...

Descriptive Statistics Singh & Thirsk (2022), Chapter 17, pp. 368-69; pp. 372-377 Salkind & Frey, 2020; Chapter 2, p.19-36; Chapter 3, p.41-53; Chapter 4, p.56-74 Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Learning Objectives State the purpose of descriptive statistics Describe measures of central tendency and their use Describe measures of variability and their use Select appropriate measures of central tendency and variability Describe a frequency distribution Understand the application and use of different charts Understand different types and shapes of distributions Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Descriptive Statistics: What is it?  A series of tests used to: – describe – synthesize – organize – discern general trends This Photo by Unknown Author is licensed under CC BY-SA-NC  Therefore, they are procedures used for describing, organizing, and interpreting information (data) Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, 3 by J. Gaudet and reviewed by Christine Houston (2024)  Ways To Describe Data  A set of data can be completely described in terms of: 1. Measures of central tendency 2. Measures of variability 3. Frequency distributions 4. Shapes of distributions Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024) 4 1. Measures of Central Tendency Measures of Central Tendency Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Discussion Questions 1. What is the formula for computing the mean? 2. How do you calculate the median? 3. How do you compute the mode? Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e 6 SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Things to Remember... The mean is sometimes represented by the letter M. n = sample size; N = population size. Sample mean is the measure of central tendency that best represents the population mean. Mean is VERY sensitive to extreme scores that can skew or distort findings. The mean is the middle point in a set of values, whereas the median is the middle point in a set of cases. Because the median cares about the number of cases, extreme scores (i.e., outliers) do not impact it. The mode is the least precise measure of CT Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  A Little About Percentiles... Used to define the percentage of cases equal to and below a certain point on a distribution. 75th percentile means that DISCUSSION QUESTION the score received is at or Why is the median always above 75% of all other at the 50th percentile rank? scores in the distribution. Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 8 2020, by J. Gaudet and reviewed by Christine Houston (2024)  When to Use What... Choosing a measure of central tendency Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Let’s Check…  According to Rawal & Yadav (2015), 25 to 40 % of patients with chronic obstructive pulmonary disease experience weight loss. A study is underway to determine the baseline average weight of patients with COPD attending an outpatient clinic. The following values represent weights (in kg) in a sample of 10 patients. 70 71 78 72 71 70 72 74 71 73 Which measure of central tendency would you report and why? Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024) 10  Summary You will almost always start by simply describing what’s there—hence the importance of understanding. the simple notion of central tendency. Next, variability, or how different scores are from one another. Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e 11 SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  2. Measures of Variability Measures of Variability Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e 12 SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024) How Distributions Can Differ in Variability and yet, all have the same mean Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024) 13  Computing the Range Range is the simplest estimate of variability. There are two types of range: – Exclusive range (most commonly used). o Range = h − l. – Inclusive range = less commonly seen) Where h is the highest score, and l is the lowest score Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e 14 SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Standard Deviation Most frequently used measure of variability. SD = s = represents the average amount of variability in a set of scores: Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e 15 SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Why n – 1? Standard deviation is an estimate of the POPULATION standard deviation. To make it an unbiased estimate, you must subtract 1 from n. This artificially inflates the SD (it makes it bigger) because it makes the denominator smaller. Computing a standard deviation Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e 16 SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Variance Variance = standard deviation squared: Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e 17 SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Things to Remember Standard deviation is computed as the average distance from the mean. Need to first compute the mean before SD. The larger the standard deviation, the more spread out the values are (the greater the variability). Like the mean, the standard deviation is sensitive to extreme scores. If s = 0, then there is no variability among scores, and the scores are essentially identical in value. Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e 18 SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Let’s Check…. For example, let’s use this data set 17 23 27 16 22 21  Compute the mean x =17+23+27+16+22+21 / 6 Why? Because you are trying to find out how much each score, on average, deviates from the mean  x = 21 Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024) 19  Now…interpret this!  Remember that the definition of standard deviation is “how much each score, on average, deviates from the mean”  For our example, the average deviation away from the mean of 21 is 4.05 What does the standard deviation of 4.05 indicate?_____________________ What then, would be the variance? ___________ Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, 20 by J. Gaudet and reviewed by Christine Houston (2024)  Summary Measures of variability helps us: – Understand distribution of data points. – Along with measures of central tendency. o Distinguish distributions. o Describe a collection of scores. o Describe what individual scores represent. Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by 21 J. Gaudet and reviewed by Christine Houston (2024)  3. Frequency Distributions Most basic method of tallying and representing the number of times a certain score occurs. Usually group scores into interval classes/ranges. A class interval is a range of numbers. Researchers can display frequency data in:  tables (Frequency Distribution Tables)  or graphically  Graphs include:  bar charts, column charts, histograms (vertical columns)  frequency polygons (dots connected by straight lines Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by 22 J. Gaudet and reviewed by Christine Houston (2024)  Frequency Distribution Example with Class Intervals Class Interval Frequency 45-49 1 40-44 2 35-39 4 30-34 8 25-29 10 20-24 10 15-19 8 10-14 4 5-9 2 0-4 1 Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by 23 J. Gaudet and reviewed by Christine Houston (2024)  Why Illustrate Data? Creating a simple chart Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e 24 SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Excerpt from: Yildiz, E. (2020). The Relationship between anxiety and intolerance of uncertainty levels in individuals who received liver transplant: A descriptive cross-sectional study. Clinical Nursing Research, 00(0), 1-10. Results Descriptive Characteristics of Participants It was determined that 36.4% of the individuals with LT studied within the scope of the research were between 44 and 56 years old (mean age = 48.34, SD = 12.66; min = 19, max = 69), 66.1% were male, 86.4% were married, 81.4% lived with their spouses and children, 42.4% had primary school education level, 82.2% had no jobs, and 80.5% had medium level economic status as expressed by themselves. It was also identified that 44.9% of the individuals with LT waited for 1 to 6 months for transplantation, 64.4% did not experience any complications in the post- transplant period, and 66.7% of those who experienced complications had infections (Table 1). Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, 25 by J. Gaudet and reviewed by Christine Houston (2024) Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by J. 26 Gaudet and reviewed by Christine Houston (2024)  Bar Graph/Chart  The graphic display of a Figure 4.6: A bar chart that frequency distribution compares different water activities  Plots the number of times a score/value occurs  Most widely used graphs for displaying nominal and ordinal level data (discrete/dichotomous data)  are distinct or separate i.e. personality type; Olympic medals  Bar graph/chart  really the same as column graph/chart Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, 27 by J. Gaudet and reviewed by Christine Houston (2024)  Histograms and Frequency Polygons  The graphic display of a frequency distribution  Plots the number of times a score/value occurs  Most widely used graphs for displaying interval and ratio level data (continuous data – in theory, have infinite number of values)  i.e. test scores; weight Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by J. 28 Gaudet and reviewed by Christine Houston (2024)  Hand-Drawn Histogram Figure 4.2: A hand-drawn histogram Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e 29 SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Frequency Polygon Figure 4.4: A hand-drawn frequency polygon Discussion Question Why use a frequency polygon rather than a histogram to represent data? Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024) 30  Pie Chart Figure 4.8: A pie chart illustrating the relative proportion of one category to Figure 4.7: Using a line chart to show a others trend over time Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by 31 J. Gaudet and reviewed by Christine Houston (2024)  4. Shapes of Distributions Examining Data - Tables and Figures Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024) 32  Normal Distribution Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 33 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Skewed Distributions Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024) 34  Let’s Check…  If the curve is negatively skewed what does this mean about the scores on a test?  Positively skewed? Going back to slide #10, interpret the skew? Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024) 35  The SPSS Output for Measures of Central Tendency Figure 2.2: Descriptive statistics from SPSS Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e 36 SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  The SPSS Output for Measures of Variability Figure 3.1: SPSS output for the variable reaction time Source: IBM Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e 37 SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Bivariate Statistics Singh & Thirsk (2022), Chapter 17, pp. 390-391 Salkind & Frey, 2020; Chapter 5, p. 76-94 Chapter 16, p. 274-286 Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by J. Gaudet and 38 reviewed by Christine Houston (2024)  Learning Objectives Describe what correlations are and how they work Describe simple correlations and their use Interpret the value of the correlation coefficient Understand what other types of correlations exist and when they should be used Understand predictions and their use Use SPSS to compute and interpret descriptive, bivariate statistics and the regression line Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 39 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Why use Bivariate Statistics? Measures of central tendency and variability are critical for describing a distribution. But… Sometimes, we are interested in describing the relationship between This Photo by Unknown Author is licensed under CC BY-SA-NC variables Bivariate correlation (two variables). Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, 40 by J. Gaudet and reviewed by Christine Houston (2024)  What Correlations Are About... rXY = correlation between X and Y. Examines the relationship between variables with values ranging from −1.00 to +1.00. How the value of one variable changes in relation to changes in another variable. Pearson product–moment correlation examines the relationship between two continuous variables (interval or ratio data) –Pearson’s r. Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e 41 SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Types of Correlation Coefficients Positive Correlation or Direct correlation Negative Correlation or Indirect Correlation Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e 42 SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Things to Remember About Correlations They range in value from _______ to _______ Absolute value indicates _________. Best to use indirect and direct, instead of ________ and __________: – Keep from assigning value to the relationship. Represented by the small letter r with a subscript representing the variables that are being correlated. Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e 43 SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  General Guideline for Interpretation Table 5.3: Interpreting a Correlation Coefficient Size of the Correlation Coefficient General Interpretation.5 to 1.0 Strong relationship.4 Moderate to strong relationship.3 Moderate relationship.2 Weak to moderate relationship 0 to.1 Weak or no relationship Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e 44 SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024) Interpreting the Correlation Coefficient The correlation coefficient is.692 – let’s say that the correlation Discussion Question coefficient was testing the What does this tell you relationship between age (IV; X) and blood pressure (DV; Y) about the relationship? Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, 45 by J. Gaudet and reviewed by Christine Houston (2024)  Other Types of Correlations Table 5.4: Correlation Coefficient Shopping, Anyone? Level of Measurement and Examples Variable X Variable Y Type of Correlation Correlation Being Computed Nominal [voting preference, such Nominal (biological sex, Phi coefficient The correlation between voting as Republican or Democrat) such as male or female) preference and sex Nominal (social class, such as Ordinal (rank in high school Rank biserial coefficient The correlation between social class high, medium, or low) graduating class) and rank in high school Nominal [family configuration, Interval (grade point Point biserial The correlation between family such as two-parent or single- average) configuration and grade point average parent) Ordinal (height converted to rank) Ordinal (weight converted to Spearman rank coefficient The correlation between height and rank) weight Interval (number of problems Interval (age in years) Pearson correlation The correlation between number of solved) coefficient problems solved and age in years Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e 46 SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Excerpt from: Orgambidez, A., Borrego, Y., & Vazquez- Aguado, O. (2020). Linking self-efficacy to quality of working life: The role of work engagement. Western Journal of Nursing Research, 42(10), 821-828 Descriptive Statistics and Correlations Table 1 presents the means, standard deviations, and correlations of all the variables. In general, the nursing professionals expressed moderated levels of self-efficacy (M = 4.51, SD = 1.03) and work engagement (M = 4.02, SD = 1.28). The participants were relatively satisfied with their jobs (M = 4.61, SD = 1.13) and affectively committed with the organization (M = 4.91, SD = 1.35). Correlation analyses showed positive and significant (p <.01) relationships between self-efficacy and work engagement (r =.43), job satisfaction (r = 0.35) and affective organizational commitment (r =.29). In the same way, work engagement was positively related to job satisfaction (r =.41, p <.01) and affective organizational commitment (r =.44, p <.01). Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, 47 by J. Gaudet and reviewed by Christine Houston (2024) Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024) 48  Visual Picture: The Scatterplot The general shape of the collection of data points indicates whether the correlation is direct (positive) or indirect (negative). Figure 5.1: A simple scattergram Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e 49 SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Strong Positive Relationships Figure 5.2: A perfect direct, or Figure 5.3: A strong, but not positive, correlation perfect, direct relationship. Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e 50 SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Strong Negative Relationship Figure 5.4: A strong, but not perfect, indirect relationship. Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e 51 SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Squaring the Correlation Coefficient Sometimes we need to go beyond stating the strength/magnitude... If we square the value of the correlation coefficient “r”, we can estimate the proportion of variability in one variable that can be determined from the relationship with the other variable The value r2 is called the ______________ __________ So… a correlation of r =.692 (or -.692), for example means that r2 = 0.478 (or 48%) of the variability in the Y scores can be predicted from the relationship with X However, 52% is unexplained. What is this called? ______ ___________________________ In other words, X (IV) tends to have an effect on Y (DV), but this is not always the case Thinking back to our age and blood pressure example…what can you say? _______________________________________________________ Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024) 52  Test Yourself ! If r=.50 for testing the relationship between exercise and quality of life scores, the coefficient of determination r2 = ___ If r = -.80 for testing the relationship between sleep and test scores, the coefficient of determination r2 = ___ If r = -.20 for testing the relationship between activity levels and height, the coefficient of nondetermination (alienation) is ___ Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024) 53  Questions A correlation coefficient comparing study time and grades is.80. Can one conclude that greater study time causes good grades? ______ The less money you put in the bank, the less interest you will earn is an example of a ______ correlation. The more oatmeal you eat the lower your blood cholesterol is an example of a _________ correlation The more you exercise, the less you will weigh is an example of a ______ correlation. The less time you take to complete a test, the more you’ll get wrong is an example of a ______ correlation. Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024) 54  How Variables Share Variance and the Resulting Correlation Figure 5.5: How variables share variance and the resulting correlation Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e 55 SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Association v/s Causality Image from: https://imgs.xkcd.com/comics/correlation.png, licensed under a Creative Commons Attribution-NonCommercial 2.5 License. Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e 56 SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Summary The correlation coefficient provides information about: – relationships between variables. – how variables change or remain the same in concert with others. o Shows how things are related. o Shows what things have in common. Useful descriptive statistic (also used in inference). Express a relationship that is associative: o But not necessarily causal. Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e 57 SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Why and How Predictions Work Prediction is the computation of future outcomes based on a knowledge of present ones When we want to predict one variable from another, we need to first compute the correlation between 2 variables We can then predict one variable from another using regression This Photo by Unknown Author is licensed under CC BY-NC-ND Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, 58 by J. Gaudet and reviewed by Christine Houston (2024)  Linear Regression Linear Regression Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by 59 J. Gaudet and reviewed by Christine Houston (2024)  Cross-Tabulation (contingency) Table A two-dimensional frequency distribution; frequencies of two variables are cross-tabulated “Cells” at intersection of rows & columns display counts and percentages Variables must be ______ or ________ Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024) 60 Contingency table for Education and Home Computers Computer * Highschool Crosstabulation Count Highschool no highschool highschool diploma diploma Total Computer home computer 42 7 49 no home computer 0 23 23 Total 42 30 72 Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e SAGE Publishing, 61 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Understanding and Interpreting the SPSS Output Figure 5.7: SPSS output for the computation of the correlation coefficient Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e 62 SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)  Scatterplot output in SPSS Figure 5.9: A simple scatterplot Adapted from Salkind, Statistics for People Who (Think They) Hate Statistics, 7e 63 SAGE Publishing, 2020, by J. Gaudet and reviewed by Christine Houston (2024)