Summary Descriptive and Inferential Statistics

Stats midterm 1
Module 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