Equations!
X = Categories of IV |
f = frequency of scores |
∑ (sigma) = sum (to add something up) |
Relative Frequency (rf) = f➗N |
N = total number of scores |
Cumulative frequency (cf) = start at bottom f and add up |
Cumulative relative frequency (crf) = cf➗N |
Range = Max # - Min # |
Population mean = μ |
Sample mean = M or x̄ |
Deviation = x - μ or x-x̄ |
Variance = Σ(x-x̄)² ፥ N |
Standard Deviation (SD) = √Variance OR √SD² |
Pearson's coefficient of skew = 3(x̄-Mdn) ➗ SD |
Types of scales of measurement!
1.) Nominal ("categories of"):
- No quantitative distinction between observations
- Categories are equivalent and discriminable: one is not better than or higher than the other(s) and can be distinguished from each other
- how many items/people are in one category/group
- do not need/include crf or cf
- Cant create stem and leaf display
2.) Ordinal ("more of"):
- the data can be categorized and ranked
- Cant create stem and leaf display
3.) Interval ("how much of"):
- the data can be categorized and ranked, and evenly spaced (e.g., temp)
- Arbitrary zero, therefore, cannot speak meaningfully about ratios
- could have negative numbers
4.) Ratio ("Proportion of"):
- Equal intervals between objects represent equal differences (Eg., money)
- Has a meaningful zero |
How we describe data
“Bell-shaped” curve |
Kurtosis |
- Normal distribution, Gaussian distribution |
- degree to which data values are distributed in the tails of the distribution |
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platykurtic distribution = low degree of peakedness (<0) |
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normal distribution = mesokurtic distribution (0) |
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leptokurtic distribution = high degree of peakedness (>0) |
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Definitions!
Descriptive statistics: Organizes, summarizes, and communicates a group of numerical observations |
Inferential statistics: Allows tests of hypotheses using systematic, objective procedures |
Discrete numbers: separate, indivisible categories (eg., 4 or 5 children, not 4.34 children) |
Continuous numbers: infinite number of values fall between any two observed values (eg., Age, height, weight, time) |
Independent variable (IV): Feature(s) of a study that is/are used to explain or explore the participants behaviour |
Dependent Variable (DV): Behaviour of the participants that we are observing, measuring, or recording |
Cumulative relative frequency (crf):proportion of scores at or below a particular score |
Cumulative frequency (cf): frequency of scores at or below a particular score |
Relative frequency (rf): fraction of the total group associated with each scores |
Modality:the number of peaks in a frequency distribution of data |
positive skew: a lot of data on the lower end of the distribution |
negative skew: a lot of data point on the higher end of the distribution |
Semi-interquartile Range (SIQR): the distance of a typical value from the median |
Median Absolute deviation (MAD): Absolute measure of how many physical units values deviate from the median |
Sum of squared deviations
1.) Compute x̄ = ∑ x X ➗N
2.) Compute the squared deviation for each score: (x−x̄)2
3.) Compute the sum of squared deviations (SS)
4.) Divide SS by N for the mean of squared deviations |
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Graphic Figures!
If you have nominal or ordinal data: use BAR GRAPH |
If you have Interval or Ratio data: use HISTOGRAM, LINE GRAPH, or POLYGON |
Measures of Central Tendency!
1.) Mode (Mod or Mo)
- most frequent category/score in a distribution
- ALWAYS a value that is observed in the dataset
- No inferential statistics
- May not be representative
2.) Median (mdn, md or x̄)
- Physical middle of an ordered set of data (aka, 50th percentile rank)
- less biased when interval/ratio data are severely skewed
- not affected by outliers or extreme scores
- No inferential statistics
3.) Mean
- Average of all numbers
- Most common value used for descriptive/inferential analyses
- Applied only to interval/ratio data
- Is biased if the scores are strongly skewed |
Data and Central Tendency!
Nominal: Mode
Ordinal: Mode, Median
Interval/Ratio: Mode, Median, Mean |
Measurement and Variance!
Nominal: none
Ordinal: range, SIQR, MAD
Interval/Ratio: Range, SIQR, MAD, variance, SD |
Interpretation of skew value
Range of Values |
Skew |
Data |
Between 0 and 0.5 |
Normal distribution |
Use Mean and SD |
Between .5 and 1.0 |
Mild to moderate skew |
Use Mean and SD |
Between 1.o and 2.0 |
moderate to strong skew |
Use Mean and SD if closer to 1.0 than 2.0 |
Greater than 2.0 |
Severe skew |
Use Median and MAD |
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Measures of Variability!
1.) Range
- Distance covered by scores in a distribution from the smallest score (min) and largest score (max)
- unreliable: sensitive to extreme values
- least preferred option of measures of variability
2.) Semi-Interquartile Range (SIGR)
- Half the range of the middle 50% of observations
- Can be used with ordinal, interval, and ratio scales
- Not affected by outliers or extreme scores
- Some values in the distribution are excluded
3.) Median Absolute Deviation (MAD)
- How to calculate it:
→ Find the median of the data set
→Compute the absolute deviation of each value in the data set from the median
→Subtract the median from the value
remove +/- (if they apply)
→Order the absolute deviation values from low to high:
→Find the median of the ordered deviation values: Mad
- less sensitive (than standard deviation) to extreme scores or skews in data
- not useful in advanced statistical procedures
4.) Variance
- average squared distance from the mean
– for computing descriptive statistics only
5.) Standard Deviation (SD)
- measure of the standard/average distance from the mean (how dispersed the scores are around the mean)
- sensitive to extreme scores or outliers and is therefore biased with skewed distributions |
Symmetrical vs. Skewed !
Symmetrical |
+ Skewed |
- skewed |
Mean and median are always the same (in the middle) |
mean the closest to the tail end |
mode varies |
mode is where the peak is |
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median is in between |
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Tail pointed towards high # |
Tail pointed towards low # |
Use Median and median absolute deviations for extremely skewed data |
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