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Summary Descriptive and Inferential Statistics
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Full summary of all modules 0-14 for year 1 statistics
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Stats midterm 1Module 0
- Variables characteristics of something or someone must have variation
If no variation constant
- Cases something or someone
Levels of measurement statistical methods
- Categorical variables each observation belongs to a set of distinct categories
Nominal different categories that vary from each other but no ranking order
Ordinal different categories and ranking order but no similar intervals
- Quantitative variables differences not too important
Interval different categories, in order AND similar intervals between categories (age)
Ratio everything + meaningful 0 point (height)
Discreet categories form a set of separate numbers (1 goal, 2 goal)
Continuous infinite region of values (height)
Module 1
Data matrix and frequency table
- Recoding for FT build ordinal categories for quantitative variables weight
- Graphs for frequency tables
Pie chart nominal/ordinal
Bar graph lots of categories
Dotplot for quantitative variables useful form small sample
Stem and leaf plot (?)
Histogram large sample bars touch each other continuous underlying scale
interval/ratio
- Histograms bell shaped UNIMODAL distribution
Symmetric
Skewed to the right (long right tail)
Skewed to the left (long left tail)
Two peaks BIMODAL distribution
Measures of central tendencies
- Mode value that occurs most frequently most common outcome nominal/ordinal
- Median middle value of observations when ordered from smallest to largest
Order from low to high pick middle result
If even n. of responses take average of two middle values
∑x
- Mean sum of all the values divided by number of observations x=
n
Nominal variables median and mean are impossible mode
In skewed distributions, the mean lies towards the direction of the skew
Outlier one data has disproportional effect on the mean
If the mean is not affected by the outlier then it is resistant
Measures of variance information on the variability/dispersion of the data
- Range highest value – lowest value only uses extreme values
- Interquartile range divides distribution in 4 leaves out extreme values
25% below and 25% above + 25% and 25% in the middle
Three quartiles Q1/ Q2/ Q3 Q2 divides in two equal parts
IQR = Q3 – Q1
Outliers if lower than Q1 – 1.5 IQR or higher than Q3 + 1.5 IQR
, - Boxplot for variability and range of data
Box = Q1+Q2+Q3 = IQR
Middle line = median = Q2
Whisker lines = other data
End of the whiskers = minimum and maximum non-outlier values
Outliers = dots
Variance and standard deviation
- Take into account all values of variable
2
∑ ( x−x )
- Variance s2=
n−1
(X – mean of X) ogni X alla seconda aggiungi tutti diviso sample size – 1
The larger the variance the larger the variability the more the values are spread
out around the mean
- Standard deviation because the variance is the metric of the variable SQUARED
average distance of an observation from the mean the larger the SD the larger the
variability of the data
√
2
∑ ( x−x )
s=
n−1
Z-scores
- Number of standard deviations removed from the mean sum of Z-scores is 0
- Good for comparison and finding potential outliers
x−x
z=
s
Positive Z-score = above the mean
Negative Z-score = below the mean
- Bell-shaped distribution the Empirical rule
68% of observation is between mean -1 and mean + 1
95% between mean -2 and mean + 2
99% between mean -3 and mean + 3
- Skewed to the right large positive z-scores (more extreme values to right)
- Skewed to the left large negative z-scores (more extreme values to left)
Regardless of shape 75% must be between +/- 2
89% between +/- 3
- Standardization recoding original scores into z-scores replace the original scores by
standard deviations from the mean common VS exceptional depends on comparison
group
Module 2
Contingency tables
- Display two categorical variables male and female x the answers
Each combination of rows is a cell showing the answer frequency then calculate
conditional proportions/percentages
Scatterplot
- Associations between continuous variables X (ind) horizontal & Y (dep) vertical
Can have positive and negative associations or none
Pearson correlation r correlation coefficient
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