Descriptive Statistics

Descriptive Statistics

1. Statistics can be found in these ways

   • Population
     • A large group of interest (e.g., all the white rats in the world form a population, as do all teenagers, all people aged 24)
     • Calculating Standard Deviation for a population
       • Formulas to find Variance and Standard Deviation:  SD*2 = ∑(X-M)*2/N-1 SD = √SD*2
       • When computing the standard deviation from a batch of scores and you want to estimate the population's standard deviation. The formula for SD changes a bit. Instead of using N you use  N- 1. Makes the SD of sample a more accurate estimate of the population.
   • Sample
     • Subset of the larger population for testing. Want to be able to generalize the sample to the population (i.e., sample should be representative of the population).
     • Calculating Standard Deviation fora sample
       • To find the standard deviation for a sample, you must go from Variance to Standard Deviation. This is the formula for calculating Variance. SD*2 = ∑(X-M)*2/N
       • X represents each score, we subtract the mean:  (X – M)
       • We square each of the resulting deviation scores: (X – M)*2   
       • We add up all the squared deviation scores: sum(X – M)*2 (This quantity is called the “sum of squares” because we sum all of the squared deviation scores. It is also represented as ∑)   
       • We divide the sum of squares by the number of scores to get the mean squared deviation, called the variance: SD*2 = ∑(X-M)*2/N  
       • This is the Formula on how to get the SD: SD =  √SD*2

2. Scales of Measurement Review

   • Nominal Scale
     • - No numerical, quantitative properties - Levels = different categories or groups
   • Ordinal Scale
     • - Order of levels from lowest to highest - Intervals between each rank are not equal
   • Interval Scale
     • - Equal intervals - Can be summarized using means - No absolute zero
   • Ratio Scale
     • - Same as interval Scale (equal intervals, summarized using means) - Has an absolute zero

3. Analyzing Research Results

   • Three approaches
     • Comparing group percentages
     • Comparing individual scores
     • Comparing group means

4. Frequency

   • How often something (a score) occurs
   • Frequency Distributions
     • Four types of graphs
       • Pie charts
       • Bar graphs
       • Frequency polygons
       • Histograms
   • Frequency Table Example

5. Central Tendency

   • A measure of what value a participants' scores tend to center on (e.g., mean, median, mode)
     • Mode
       • Mode is the most common score in the distribution. The score that the most participants got in a distribution.
     • Median
       • Median (middle score in distribution of scores) is when the scores are lined up in distribution (lowest to highest) and the median will be the middle score.
       • Example: 61, 72, 75, 81, 94 The median will be 75 since it is right in the middle of the distribution.
       • If there isn't a middle score due to the number of scores adding up to a even number like this: 58, 60, 61, 72, 73, 75, 81, 84, 89, 94 Then the median score will be the two scores that are closer to the middle 73+75/2=74, so the Median is 74 
     • Mean
       • Mean (average) is adding all the individual scores in the batch and then dividing by the number of scores
       • Formula for Mean M =∑X/ N - M = mean - X = individual's score on a test - ∑ = the sum of (add all the values) - N = number of scores

6. Distribution of Scores

   • Normal Distribution
     • Standard Deviation: computation that quantifies how far (on average) the individual scores in the data set are from the mean
     • Variance: Computation that describes how spread out the scores of a sample are around the mean. The squared value of the standard deviation.
   • Skewed Distribution
     • Usually has cluster of scores at one end of a distribution and a long tail of scores trailing off to right or left of distribution. Best visualized by a histogram.
     • 3 types of Distribution skews
       • Negative
         • Negative skew tail points towards negative scores (or lower scores/towards left) of a distribution's peak
         • This makes the Mean to be on the left of the peak
       • Normal (no skew)
         • This refers to the tails on either side of the peak are symmetrical; making the Mean to be at the peak of the distribution.
       • Positive
         • The tail points towards positive scores (or greater scores of a distribution's peak. 
         • The Mean is to be at the right of the peak.

7. Effect Size

   • Effect size provides a scale of values that is consistent across all study types.
   • Describing Effect Size
     • When a study's outcome demonstrates a difference between two groups, the effect size describes how large or small that difference is.
     • Calculation describing the strength of the association between two variables. Example: Pearson's r, Cohen's d
     • Pearson's r
       • Pearson's r is usually used in correlations. Here is some ranges in how effects are correlated with r - Small effect near r = .15 - median effect near r = .30 - large effect near r = .40 These correlations describe the strength and direction of the relationship. Positive (+) direction is going right while negative (-) direction is going left. Values ranges from 0.00 to ±1.00
       • r*2 (squared value of Pearson's r) is able to change the value of r to a percentage. This is refer to a percentage of shared variance between two variables Formula: r*2 = (r x r) 100
     • Cohen's d
       • Cohen's d is used in experiments that have two or more experimental conditions
       • It expresses the size of the effect of the Independent and dependent variables.
       • Here are some size of effect in terms of SD units: d = 1.0 means 1 SD apart d = .2 means .2 SD apart

8. z-scores

   • z-scores are used to describe where the individual scores stand in relation to an entire set of scores.
   • z-score describe if a score is above or below the mean, and how far from the mean it is (it is measured in SD units).
   • Computing z-score
     • The formula to compute z-score: z = (X-M)/SD

9. Bivariate Correlations

   • Association between two measured variables
     • The two variables can either be Quantitative or Categorical variables. These variables that are branched out from here are graphed differently
     • Quantitative Variables
     • Categorical variables

10. Important Considerations

    • Restriction of the range issues
      • - Magnitude of correlation coefficient is reduced - Retricted range = retricted variability in scores (therefore less variability can be explained) - When participants in sample respond very similarly on variable of interest (e.g., limited age range)
    • Linearity
      • - Pearson's correlation only detects linear relationships (not curvilinear relationships).
    • Importance of plotting data