Statistics And Correlation

Statistics / Correlation research

After a research project has been carried out, what are the results? For quantitative data, the results are a bunch of numbers. Now what? What do the numbers look like, what do the numbers mean Statistical analysis allows us to: Summarize the data Represent the data in meaningful ways Determine whether our data is meaningful or not

Many forms of research Many forms of data Variety of dependent variables Data can take 1 of 4 different forms. Four measurement scales: Nominal Ordinal Interval Ratio

Nominal scale – simplest form of measurement: you give something a name. Qualitative scale of measurement Assign participants to a category based on a physical or psychological characteristic rather than a numerical score. E.g., Male vs. Female; color of eyes Intelligence levels: smart vs. dull Data is determined by a strict category Only allows for crude comparisons of results. Can really only be used for qualitative comparisons.

Ordinal scales ranking system – data is ranked from highest to lowest Show relative rankings but say nothing about the extent of the differences between the rankings. Does not assume that the intervals between rankings are equal. E.g., rank 10 smartest kids E.g., college football rankings Problem – no absolute magnitude Makes it difficult to make comparisons

Interval scales – numeric scores without absolute zero Not only relative ranks of scores, but also equal distances or degrees between the scores. Interval = equal intervals ordering E.g., IQ scores – difference between 100 and 120 is the same as the difference between 60 and 80. Problem – no absolute zero Cannot have an IQ score of 0. Does not allow for ratio comparisons. E.g., IQ of 120 is not twice as smart as 60.

Ratio scales - numeric scores but with an absolute zero point All of the properties of the other scales but with a meaningful zero point. Allows you to make ratio comparisons i.e., is one twice as much as another? E.g., number of correct answers on an exam. E.g., number of friends a person has.

Nominal and Ordinal scales are discrete or categorical Interval and Ratio scales are continuous scales. NOIR  Increasing levels of resolution Most observable behaviors are measured on a Ratio scale. Most psychological constructs are measured on an Interval scale. Important to recognize what scale of measurement is being used. Nominal and ordinal data require different statistical analyses than interval or ratio data.

After data collection is finished, the data must be summarized. What does it look like? Start with exploring the data. Look at individual scores. Frequency distributions show us the collection of individual scores. Simple frequency distributions – lists all possible score values and then indicates their frequency. Allows us to make sense of the individual scores.

Grouped frequency distribution – raw data are combined into equal sized groups

Histogram – a frequency distribution in graphical form Bar graph

Numeric summaries that condense information Numbers that are used to make comparisons Numbers that portray relationships or associations. Two main types of stats Descriptive statistics Inferential statistics

Descriptive statistics – summarize results Central tendency Variability Inferential statistics – Used to determine whether relationships or differences between samples are statistically significant

Central tendency – what is the “heart of the data”? Three measures of central tendency Mean – average Add up all scores and divide by the total number of samples Median – middle score Line up all scores and find the middle one Mode – most common score Which score occurs the most often

Simply add up all of the scores and divide by the number in the sample. The statistic for a sample – X bar - =  X / n

=  X / n  = 1822 / 23 = 79.22

Pros and cons of using the mean Pros Summarizes data in a way that is easy to understand. Uses all the data Used in many statistical applications Cons Affected by extreme values E.g., If Robert would have scored a 0, the mean changes to 74. E.g., average salary at a company 12,000; 12,000; 12,000; 12,000; 12,000; 12,000; 12,000; 12,000; 12,000; 12,000; 20,000; 390,000 Mean = $44, 167

Median – the middle score in the data: half the scores are above it, half of the scores are below it. Scores are ranked…. Find the one in middle. 50 56 66 68 70 72 76 76 76 78 78 78 78 80 80 86 86 86 88 96 98 100 100 Example – Median is the score 78. If there is an even number of scores, the median is the average of the two middle scores. E.g., 10, 10, 9, 9 – Median is 9.5

Pros and cons of using the median Pros Not affected by extreme values Always exists Easy to compute Cons Doesn't use all of the data values Categories must be properly ordered Mean is almost always preferred. Exception: data is skewed, not distributed symmetically, or has extreme scores.

Mode – the most common score of the data Mode is 78

Pros and cons of using mode Pros Fairly easy to compute Not affected by extreme values Cons Sometimes not very descriptive of the data Not necessarily unique – if two modes = bimodal; if multiple modes = polymodal. Doesn't use all values.

Variability – how spread out is the data Measures of variability Range Variance Standard deviation – “average variability” Range – the simplest variability statistic = high score – low score. Standard deviation - a measure of the variation, or spread, of individual measurements; a measurement which indicates how far away from the middle the scores are.

The larger the standard deviation, the more spread out the scores are. The smaller the standard deviation, the closer the scores are to the mean.

Computing SD 1. subtract each score from the mean Ex. (100 – 80 = 20) 2. square that number for each score 3. add up the squared numbers. This is the “sum of squares” 4. Divide the sum of squares by the total number in the sample minus one - this is the variance 4. take the square root of that number. This is the standard deviation

Data is usually spread around the mean in both directions Some are higher than the mean, some are lower. The frequency distribution of the scores tells us how the scores land relative to the mean. Ideally, some scores are higher, some are lower, most are in the middle. The normal distribution – the bell curve

As sample size increases, the distribution of the data becomes more normalized. Importance of the normal distribution Symmetrical Mean, median, mode all the same The further away from the mean, the less likely the score is to occur Probabilities can be calculated

We can assume that many human traits or behavior follow the normal distribution Some are high is a trait, some are low, but most people are in the middle. E.g., personality traits, memory ability, musical capabilities People have a tendency to think categorically - erroneous

All data points are arranged, and a particular data point is compared to the population. E.g. IQ score of 130 Percentile reflect the percentage of scores that were below your data point of interest. IQ score of 130 is at the 95 th percentile. Percentile is arranged according to standard deviation.

0 SD is the 50 th percentile 1 SD is the 84 th percentile 2 SDs is the 97 th percentile 3 SDs is the 99.5 th percentile

Advanced statistics that reveal whether differences are meaningful. Take into account both central tendency (usually the mean) and variability Determines the probability that the differences arose due to chance. If the probability that the observed differences are due to chance is very low, we say that the difference is statistically significant. Science holds a strict criteria for determining significance.

α = alpha – the probability of committing a Type I error. α is normally set at 0.05. Only a 5% chance of committing a type I error. Can find the probability that the observed differences are statistically significant. If that probability is less than 0.05, the results are statistically significant. Many types of inferential statistics t test Analysis of Variance

Visually representing the data can make it more understandable for you as well as anyone else looking at your results. Horizontal axis is the X-axis Vertical axis is the Y-axis The best graph is the one that makes the data more clear.

Each score is divided into two parts, a stem and a leaf The leaf is the last digit of the score The stem is the remaining digit(s) E.g., 49 would have 4 as the stem and 9 as the leaf. Graphing a stem and leaf is like making a table.

Stem Leaf 5 6 6 068 7 0266688888 8 006668 9 68 10 00

Much of the time, a plot of the means is useful.

Line graphs are especially important for Repeated Measures

Show the median and distribution of scores. Also shows outliers – scores that are more than 3 standard deviations from the mean.

Keys to making figures: Keep it simple Nothing is “required” for making figures Purpose is to better illustrate the results. Don’t “lie” with figures. Axes should be set at appropriate range.

Correlational research investigates the relationships between two variables. E.g., is there a relationship between poverty levels and crime Attachment level in children and future behavior. Are the number of hours husbands spend watching sports associated with wives’ marital satisfaction? Are basketball players heights associated with number of points scored?

Establishes the relationship between the variables Whether it exists The strength of the relationship Correlation can be used as a method for conducting research, or as a tool within the research.

Correlation does not mean causation Ex. Significant correlation between ice cream sales and murder rates – ice cream sales and shark attacks The number of cavities in elementary school children and vocabulary size have a strong positive correlation. Skirt lengths and stock prices are highly correlated (as stock prices go up, skirt lengths get shorter).

Can be causation, but correlational research is not designed to assess that. Meanings of correlation: Causation: Changes in X cause changes in Y Common Response: changes in X and Y are both caused by some unobserved variable. Confounding variables are causing Y and not X .

Correlation simply measure relationships. All methods use to calculate correlation are established so that it can vary between –1 and +1. Most common method is the Pearson product-moment correlation coefficient Represented by r Strength of the correlation The closer to +1 or -1, stronger the correlation

Positive correlations – as X increases, Y increases. Ex. Horsepower and speed The value of the correlation represents the strength of the relationship. +1 represents a perfect positive relationship. 0.9 is an extremely high correlation, 0.2 isn’t as strong. Zero correlations – as X increases, we have no idea what happens to Y. Values around 0 Examples: length of hair and test scores

Negative correlations – as X increases, Y decreases. Horsepower and miles per gallon Important: a negative correlation simply tells what direction the relationship is, not the strength of the relationship. One way to view correlations is graphically. Scatterplots – graph that plots pairs of scores: one variable on the X axis, one on the Y axis.

Concurrent Change Same Direction

Negative Correlation Concurrent Change in Opposite Directions

Scatter plots also allow you to see outliers. Most correlations are assessing a linear relationship. Some relationships are more complex. E.g., the Yerkes-Dodson law

Statistics And Correlation

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Statistics And Correlation