Descriptive Statistics

Descriptive Statistics

1. Levels of Measurement

   • nominal
     • attributes only named
   • ordinal
     • attributes can be ordered
   • interval
     • distance is meaningful
   • ratio
     • absolute zero

2. Frequency

   • state of occurrences a value appears in a certain category 
   • proportion
     • divide the frequency by the total of all the frequencies
   • cumulative
     • sum of the frequency and the frequencies of all the preceding frequencies which are listed in some sensible order
   • cumulative proportion
     • the sum of the proportion frequency and the proportion frequency of all of the preceding frequencies which are listed in some sensible order
   • percentage
     • the number per hundred who have a certain characteristic

3. Displays

   • table
   • graph
   • chart

4. Distributions

   • Skewed
     • positive
     • negative
   • Normal
   • Bimodal
   • multimodal

5. Central Tendency

   • Mean
     • Such that it is the balance point of the data
   • median
     • Such that it lies at the midpoint of the distribution
   • mode
     • Such that it is the most frequently occurring score
   • parameter
     • a numeric quantity, usually unknown, that describes a certain population characteristic
   • statistic
     • a quantity, calculated from a sample of data, used to estimate a parameter.

6. Measures of variability

   • range
     • Such that it is the measure of the distance from the lowest and highest cases
   • interquartile range
     • Such that it is the measure of the range of the middle 50% of the cases
   • variance
     • Such that it is the measure of the average squared deviation from the mean
   • standard deviation
     • Such that it is the positive square root of the Variance.  (S.D., s, σ)

7. Standard scores

   • z-score
     • Such that it centers the mean at 0 by determining how many standard deviation units a subject’s  score is from the mean and whether the score is above or below the mean
   • t-score
     • Such that it centers the mean at 50 by multiplying each subject’s z-score by 10 and adding 50
   • CEEB, SAT, GRE
     • Such that it centers the mean at 500 by multiplying each subject’s z-score by 100 and adding 500

8. Correlations

   • data
     • parametric
       • statistical tests assume that the data are “normally distributed
     • non-parametric
       • ■statistical tests don’t make an assumption about the distribution of data.
   • relationships
     • direct/positive
       • Scores high on one variable tend to score high on the other, and scores low on one variable tend to score low on the other
     • indirect/negative
       • Scores high on one variable tend to score low on the other
   • Pearson r
     • ■The magnitude is the strength of the correlation. ■The closer the correlation is to either +1 or -1, the stronger the correlation. ■If the correlation is 0 or very close to zero, there is no association between the two variables.
   • Coefficient of determination, r2
     • §The correlation coefficient is the basis of the coefficient of determination, which tells the proportion of variance that two variables in a bivariate distribution have in common. 
   • regression
     • is a method that is used to predict scores for one variable when you have measurements on a second variable
   • Spearman's rho

9. Hypotheses

   • Null
     • there is no expected effect on the dependent variable due to the independent variable
     • Error
       • type I
         • When an alpha of .05 is used, researchers are, in effect, willing to be wrong 5 times in 100 in rejecting the null
       • type II
         • A Type II error is the error of failing to reject the null when it is false.  This type of error can have serious consequences
     • one sample Z-test
       • Z-test = sample mean – population mean / SEM.  When one sample is available, can be used to determine whether the null hypothesis is viable
     • t-test
       • ¡most often used to test the null hypothesis that the means of two groups are the same. In comparing two means, the researcher uses the two sample means, the group variances, and the sample size with a formula that generates a number, called the t value or t statisticused to compare the means of two samples for statistical significance.
   • Research
     • a definite statement that there is a relationship between variables
     • directional
       • States which Independent Variable (IV) treatment is expected to result in the better or worse effect on the Dependent Variable (DV). 
     • non-directional
       • States that the IV is expected to have an effect on the DV, but it does not specify which IV treatment will have the better or worse effect on the DV.
   • p-value
     • the likelihood of what you found in your sample would be found in the population, if the null hypothesis is true

10. Central Limit Theorem

    • The tendency of the distribution of sample means to be normal regardless of the shape of the underlying distribution
    • population
      • the entire group a study seeks to base a hypothesis on
    • sample
      • a finite selection taken from the population
    • SEM
      •  The standard deviation of a distribution of any summary statistic
      • The standard deviation of a distribution of any summary statistic
      • point estimate
        • a specific estimate of a parameter
      • confidence interval
        • interval estimates, which are attempts to set limits that have a high probability of encompassing the true (population) value of the mean ( the mean of a whole population of observations)