Data Types, Scales, and Distributions

• Applied statistics is the main focus
• Statistics are useful for testing research questions

Statistical Questions

• Translating a research question into a statistical question facilitates analysis
• A research question leads to a statistical question
• Example: Having 40 participants drive in a simulator while texting, and another 40 in the same simulator without texting
• Record the number of lane deviations for each driver
• Compute average lane deviation for each group
• Texting group vs no cell phone group
• Driving simulator provides a controlled lab environment

Independent and Dependent Variables

• Independent variable (IV): manipulated by the researcher (texting)
• Dependent variable (DV): outcome measured (lane deviations)
• IV influencing DV: Texting increases lane deviations

Confounding Variables

• Aim to isolate the impact of the manipulated variable on dependent variable
• Consider other factors influencing lane deviations
• Age and driving experience
• Road difficulty (winding vs straight)
• To deal with confounding factors, hold variables constant by keeping them constant
• Match variables: similar age/weather conditions
• Random assignment may address confounding variables

Analysis of Statistical Questions

• Turn research question into a statistical question
• Example: Is the average lane deviation greater for the texting group than the no cell phone group?
• Apply appropriate statistical procedures, such as a 2 sample t-test

Conclusion

• Statistical conclusion has to be in the form Research Conclusion
• The average number of lane deviations is greater for texting group versus the no cell phone group is an example of statistical conclusion
• Texting while driving tends to create unsafe roads is an example of research conclusion

The overall Process

• Research Question > Statistical Question > Data Collection & Analysis > Statistical Conclusion > Research Conclusion

Population vs Sample

• Population: all observations an investigator wishes to draw conclusions about
• Population described by mean (μ) and variance (σ²)
• Sample: a subset of the population used for analysis
• Sample described by mean (y-bar/x-bar) and variance (s²)
• Parameter is population
• Statistic is sample

Statistics

• Descriptive Statistics: used to organize and summarize observations
• Inferential Statistics: use statistics from a sample to draw conclusions about the population
• Large, random samples yield statistics approximate the population characteristics

Other Terminology

• Variable vs. Constant
• Discrete or Continuous data
• Qualitative or Quantitative data

Synonyms

• Independent variable & factor are often used interchangeably in experiments
• Predictor & explanatory variable can replace the independent variable in non-experiments
• Dependent variable = outcome, response, criterion

Levels of Measurement

• Measurement: assigning numbers to observations, must be labels
• Nominal: mutually exclusive & collectively exhaustive (MECE) labels
• Assign label unique to the person, object, or event
• MECE: once a person is in one label, they can't be in another
• Example: female(1), male(2)
• Ordinal: Labels still MECE, but also indicates order of magnitude
• Example: A, B, C, D, F, or Freshman, Sophomore, etc., or ranked 1st, 2nd, 3rd
• Interval: Labels still MECE and also indicate order of magnitude and Equal intervals
• Equal differences between numbers reflect equal magnitude differences between the corresponding classes.
• The Fahrenheit scale can refer to equal differences between 80F & 40F vs. 60F & 20F, But cannot unambiguously say that 80F is twice as hot as40F. The 0 point on the Fahrenheit scale is not a true zero point
• Ratio: Still MECE, order of magnitude, & equal intervals, and there is an absolute 0 point
• The ratio between measurements has meaning
• Example: Kelvin scale, height, weight
• Region of the Country: would be nominal data

Frequency Distributions

• Quantitative/qualitative data
• Can be grouped (range of #s ex. 20-30 w/midpoint) or ungrouped (each row is assigned one number, ex. 21,22,23, etc.)
• Histogram shape is frequency table/data

Stem and Leaf Plot

• Quantitative data, retains original data
• Leaves are the last significant digit
• Stems are the remaining digits
• To correctly interpret, check the key
• Ex. 6/8 means 68; 1/7 means 1.7, we don't lose any info

Shapes of Frequency Distributions

• J-shaped: majority of data piled up towards the end of the scale
• Positively skewed: hump towards the left, more data at the lower end of the scale
• Negatively skewed: hump towards the right, more data at the higher end of the scale
• Rectangular distribution
• Bimodal: 2 major groups of data: one on the lower end and the other on the higher end
• Bell-shaped: data piled in the middle

Central Tendency

• Measures of Central Tendency: represent the center value of observations
• Mode (Mo): score/data w/ most frequency and can have more than one. Can also be used with qualitative data
• Median (Mdn): point in (ordered) distribution of scores that divides data into two groups having equal frequencies. Is only sensitive to the number of scores above & below it. Tends to be used to represent center for positively or negatively skewed data.
• Arithmetic mean: the sum of the scores divided by the total number of scores, is very stable and sensitive to extreme scores.
• Balance point of a distribution: fulcrum in the middle of the data so that the seesaw will be in perfect balance

Score Transformations

• If add a constant number to each score in a distribution, the distribution shifts by the amount of the constant, and so the mean will shift by the same amount.
• If we multiple (or divide) each score in a distribution by a constant, the mean will be multiplied (or divided) by the same constant

Variability

• Do the scores in a distribution cluster around a central point or do the scores spread around it?
• Measures of variability:
• Range: difference between highest score and lowest score, crude measure of variability depends only on two scores.
• Semi-interquartile range: half of the middle 50% of the scores Q = (Q3-Q1)/2, less sensitive to extreme scores
• Variance
• Standard deviation

Deviation Scores

• We need for other measures of variability
• For a distribution of scores, calculate the mean, then subtract the mean from each score
• These are the scores show how far a given score is from a central point.
• Variance of population: Average Squared Deviation from the Mean – N = whole population and Unbiased variance of a sample - n = sample of population
• Standard deviation of population: Average Deviation from the Mean and Unbiased standard deviation of a sample. It is the variation's square root-variation is deviation is squared

Score transformations

• Adding a constant number to each score in a distribution does not affect any of the measures of variability

If we multiple or divide each score in a distribution by a constant:

• The standard deviation will change by multiplying (or dividing) by absolute value of the constant.
• The variance will change by multiplying or dividing by the squared constant

Statistical Reasoning

• Can the variance be a negative number? (no)
• Can the standard deviation be a negative number? (no)
• Think of what the distribution of scores would look like if all the scores clustered around the mean vs spread around the mean.

Standard Scores & normal curve

• Your raw score on a test was 346
• Raw scores not very informative
• Need a frame of reference
• Use the Mean and standard deviation
• Now, we can state the position of a raw score relative to the mean in standard deviation units
• In a population use Z = Υ-μγ /σy and in a score =Y-Y/Sy
• When we convert raw scores to z scores, the mean of z scores equal 0 and standard deviation must equal 1.
• By necessity, variance must also equal !

The shape of the new distribution

• The shape distribution will not differ from the original distribution
• Sign of z score tells us something useful, above (+) or below (-) the mean
• Absolute value of z tells us the distance between the score and the mean in standard deviation units

Kinds of Standard scores

• California Psych Inventory
• Standard-Binet Intell Scale
• Wechsler Intell Scale

Given a Z score

• We can convert a set of scores to have any mean or standard deviation we would like
• Y = Y + z(Sy)
• If we divide this distribution it should approximately follow the distribution.
• The of individuals above or below a score:
• The (proportion) of individuals or of scores
• Area under any normal curve sums to 1.0
• We will focus on the standard normal curve

Example of applying a normal curve

• Cognitive ability scores follow a normal curve with μ = 100 & σ = 15
• To know the proportion of individuals with an ability score greater than 130.
• Convert raw score to z score (z = 2)
• Z = Υ-Ỹ(μ)/(sy) = (130-100) / 15 = 2
• Find z = 2 in Table A (Appendix D). Look under the column labeled “AREA BEYOND z." which is.0228

SAT Score example

• Follows a normal distribution with μ = 500 & σ = 100
• To know a student needs to the students need to have In order to be in the top 15% of the SAT distribution.
• Partition the standard normal curve such that 15% of the distribution is to the right of particular z & 85% is to the left of the same z
• In Table A, look under the column labeled “AREA BEYOND z” to get as close as possible to 15% (.15) it equal z = 1.04
• Then, Y = 500 + (1.04)(100) which equals 604

Correlation

• Measures the linear relationship between two variables
• Requires pairs of scores for each participant
• Values range from -1.0 to 1.0.
• Sign indicates whether correlation is positive or negative, which is attributed to Pearson
• Absolute value indicates the degree of linear relationship
• Correlation between 0.65 and -0.65 have same degree of linear relationship only are positive
• Weaker degree of correlation are 0.79 or -0.85 is stronger because it is closer to –1 than the other is to +1

Cohen's conventions (ignore sign)

• .1(small) correlation
• 3 (medium) correlation
• .5 (large) correlation
• predict the relation between :
• height and shoe size is positive
• cholesterol/Intelligence is none
• hours of exercise and body type is negative
• fruits and vegetables or risk of heart disease is also negative

Correlation Equation- Equation 7.3

or= Σ(x−x)(y−y)/ √SSX SSY

• SSx is sum of squares for x
• SSy is sum of square for y
• SCP is sum of cross square

Correlation example

• Administer quiz (X) before Exam 1, the obtain exam 1 scores (Y)
• Use Equation to determine results = stronger results

Spearman order ranking-order

• Measure Pearson Ranking
• 3 step process

1. Have x be ranked
2. Have y be ranked
3. New Ranked Data rs

Ranked Data

• If there repeats use this code:
• Give a number, add total ranks divided by repeated value, that answer new rank
• Ice cream sales and aggravated assaults are correlated, with causes Season
• Satification job is a performance by causes Both
• Measure with linear relationship
• Effects a range of talent or restriction
• Matrix for each individual
• Can have variables p(p-1)2 in corner

(Simple Linear) regression

• Focus more on one group
• Based on mood can perdict some actions
• Find line of beast fit with equations

Notation

• X used to represent actual scores on X
• Y used to represent actual scores on Y
• y' used to represent predicted scores on y
• y-y equation

Least squared

• When sum all squared discrepancies we prefer the value that comes out to become some possible sum
• Equation= Y'=bx-a and Y regreased on
• B=- r score
• Y’bx
• Regression helps "plug in values" to help estimate/evaluate
• In other words, use scores on X that are:
• Is the mean of the z scores for the x values or y values?

Homoscedasticity

• The spread of the variables in is the same and around 0

Estimation

• Can come to an 0 or 1 to help identify the score

Proportion of Explained Variance

• Coefficient of Determination: the proportion of variance in is determined by
• ssy1/ss
• If its high that score will likely have that relation

This quiz explores different data types like ordinal, interval, and ratio, emphasizing appropriate scales and distributions. Understand data analysis and visualization techniques for economics and statistics. Test your expertise now!