subject.id age sex
1 11 24 m
2 2 31 m
3 31 17 fDescriptive statistics
Quantitative and graphical techniques for summarizing data
Today’s agenda
- [lecture] frequency distributions, measures of spread and center, measures of association
- [lab] descriptive statistics and simple graphics in R
Last time
Data semantics
Data are a collection of measurements taken on a sample of study units:
- measured attributes are called variables
- per-unit measurements are called observations
Variables are classified by their values:
- categorical data: ordinal (ordered) or nominal (unordered) categories
- numeric data: continuous (no ‘gaps’) or discrete (‘gaps’) numbers
Dataset: FAMuSS study
Observational study of 595 individuals comparing change in arm strength before and after resistance training between genotypes for a region of interest on the ACTN3 gene.
Pescatello, L. S., et al. (2013). Highlights from the functional single nucleotide polymorphisms associated with human muscle size and strength or FAMuSS study. BioMed research international.
| ndrm.ch | drm.ch | sex | age | race | height | weight | genotype | bmi |
|---|---|---|---|---|---|---|---|---|
| 40 | 40 | Female | 27 | Caucasian | 65 | 199 | CC | 33.11 |
| 25 | 0 | Male | 36 | Caucasian | 71.7 | 189 | CT | 25.84 |
| 40 | 0 | Female | 24 | Caucasian | 65 | 134 | CT | 22.3 |
| 125 | 0 | Female | 40 | Caucasian | 68 | 171 | CT | 26 |
Descriptive statistics
Descriptive statistics refers to analysis of sample characteristics using summary statistics (functions of data) and/or graphics.
For example:
| genotype | avg.change.strength | n.obs |
|---|---|---|
| TT | 58.08 | 161 |
| CT | 53.25 | 261 |
| CC | 48.89 | 173 |
We call these descriptions and not inferences because they describe the sample:
Among study participants, those with genotype TT (n = 161) had the greatest average change in nondominant arm strength (58.08%).
The appropriate type of data summary depends on the variable type(s)
Categorical frequency distributions
For categorical variables, the frequency distribution is simply an observation count by category. For example:
| participant.id | genotype |
|---|---|
| 494 | TT |
| 510 | TT |
| 216 | CT |
| 19 | TT |
| 278 | CT |
| 86 | TT |
| CC | CT | TT |
|---|---|---|
| 173 | 261 | 161 |
Numeric frequency distributions
Frequency distributions of numeric variables are observation counts by “bins”: small intervals of a fixed width.
A plot of a numeric frequency distribution is called a histogram.
| participant.id | bmi |
|---|---|
| 194 | 22.3 |
| 141 | 20.76 |
| 313 | 23.48 |
| 522 | 29.29 |
| 504 | 42.28 |
| 273 | 20.34 |
| (10,20] | (20,30] | (30,40] | (40,50] |
|---|---|---|---|
| 69 | 461 | 58 | 7 |
Histograms and binning
Binning has a big effect on the visual impression. Which one captures the shape best?
Shapes
For numeric variables, the histogram reveals the shape of the distribution:
- symmetric if it shows left-right symmetry about a central value
- skewed if it stretches farther in one direction from a central value
Modes
Histograms also reveal the number of modes or local peaks of frequency distributions.
- uniform if there are zero peaks
- unimodal if there is one peak
- bimodal if there are two peaks
- multimodal if there are two or more peaks
Your turn: characterizing distributions
Consider four variables from the FAMuSS study. Describe the shape and modes.
Your turn: characterizing distributions
Here are some made-up data. Describe the shape and modes.
Common statistics as measures
Most common statistics measure a particular feature of the frequency distribution, typically either location/center or spread/variability.
Measures of center:
- mean
- median
- mode
Measures of location:
- percentiles/quantiles
Measures of spread:
- range (min and max)
- interquartile range
- variance
- standard deviation
The most appropriate choice of statistic(s) depends on the shape of the frequency distribution.
Measures of center
There are three common measures of center, each of which corresponds to a slightly different meaning of “typical”:
| Measure | Definition |
|---|---|
| Mode | Most frequent value |
| Mean | Average value |
| Median | Middle value |
Suppose your data consisted of the following observations of age in years:
19, 19, 21, 25 and 31
- the mode or most frequent value is 19
- the median or middle value is 21
- the mean or average value is 19+19+21+25+315 = 23
Comparing measures of center
Each statistic is a little different, but often they roughly agree; for example, all are between 20 and 25, which seems to capture the typical BMI well enough.
The less symmetric the distribution, the less these measures agree.
Robustness to skew
The mean is more sensitive than the median to skewness:
Comparing means and medians captures information about skewness present since:
- right skew: mean > median
- left skew: mean < median
- symmetry: mean ≈ median
For skewed distributions, the median is a more robust measure of center.
Percentiles
A percentile is a threshold value that divides the observations into specific percentages.
Percentiles are defined by the percentage of data below the threshold, for example:
- 20th percentile: value exceeding exactly 20% of observations
- 60th percentile: value exceeding exactly 60% of observations
Sample percentiles are not unique!
| age | 19 | 20 | 21 | 25 | 31 |
| rank | 1 | 2 | 3 | 4 | 5 |
Any number between 19 and 20 is a 20th percentile since it would satisfy:
- 20% below (19)
- 80% above (20, 21, 25, 31)
Usually, pick the midpoint: 19.5.
Cumulative frequency distribution
The cumulative frequency distribution is a data summary showing percentiles. Think of it as percentile (y) against value (x).
Interpretation of some specific values:
- about 40% of the subjects are 20 or younger
- about 80% of the subjects are 24 or younger
Your turn:
- Roughly what percentage of subjects are 22 or younger?
- About what age is the 10th percentile?
Common percentiles
The five-number summary is a collection of five percentiles that succinctly describe the frequency distribution:
| Statistic name | Meaning |
|---|---|
| minimum | 0th percentile |
| first quartile | 25th percentile |
| median | 50th percentile |
| third quartile | 75th percentile |
| maximum | 100th percentile |
Boxplots provide a graphical display of the five-number summary.
Boxplots vs. histograms
Notice how the two displays align, and also how they differ. The histogram shows shape in greater detail, but the boxplot is much more compact.
Measures of spread
The spread of observations refers to how concentrated or diffuse the values are.
Two ways to understand and measure spread:
- ranges of values capturing much of the distribution
- deviations of values from a central value
Range-based measures of spread
A simple way to understand and measure spread is based on ranges. Consider more ages, sorted and ranked:
| age | 16 | 18 | 19 | 20 | 21 | 22 | 25 | 26 | 28 | 29 | 30 | 34 |
| rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
The range is the minimum and maximum values: range=(min,max)=(16,34)
The interquartile range (IQR) is the difference [75th percentile] - [25th percentile] IQR=29−19=10
Deviation-based measures of spread
Another way is based on deviations from a central value. Continuing the example, the mean age is is 24. The deviations of each observation from the mean are:
| age | 16 | 18 | 19 | 20 | 21 | 22 | 25 | 26 | 28 | 29 | 30 | 34 |
| deviation | -8 | -6 | -5 | -4 | -3 | -2 | 1 | 2 | 4 | 5 | 6 | 10 |
The variance is the average squared deviation from the mean (but divided by one less than the sample size): (−8)2+(−6)2+(−5)2+(−4)2+(−3)2+(−2)2+(1)2+(2)2+(4)2+(5)2+(6)2+(10)212−1
In mathematical notation: S2x=1n−1n∑i=1(xi−ˉx)2
Deviation-based measures of spread
Another way is based on deviations from a central value. Continuing the example, the mean age is is 24. The deviations of each observation from the mean are:
| age | 16 | 18 | 19 | 20 | 21 | 22 | 25 | 26 | 28 | 29 | 30 | 34 |
| deviation | -8 | -6 | -5 | -4 | -3 | -2 | 1 | 2 | 4 | 5 | 6 | 10 |
The standard deviation is the square root of the variance: √(−8)2+(−6)2+(−5)2+(−4)2+(−3)2+(−2)2+(1)2+(2)2+(4)2+(5)2+(6)2+(10)212−1
In mathematical notation: S2y=√1n−1n∑i=1(xi−ˉx)2
Interpretations
Listed from largest to smallest, here are each of the measures of spread for the 12 ages:
| min | max | iqr | variance | st.dev | avg.dev |
|---|---|---|---|---|---|
| 16 | 34 | 8.5 | 30.55 | 5.527 | 4.667 |
The interpretations differ between these statistics:
- [range] all of the participants are between 16 and 34 years old
- [IQR] the middle 50% of participants are within 8.5 years of age of each other
- [variance] participants’ ages vary by an average of 30.55 squared years
- [standard deviation] participants’ ages vary by an average of 5.53 years
Robustness to outliers
Percentile-based measures of location and spread are less sensitive to outliers
Consider adding an observation of 94 to our 12 ages. (This is called an outlier.)
The effect of this outlier on each statistic is:
- IQR increases by 5.88%
- SD increases by 264.09%
In the presence of outliers, IQR is a more robust measure of spread.
Choosing appropriate measures
To determine which measures of spread and center to use, simply visualize the distribution and check for skewness and outliers.
- strong skew or large outliers: prefer median/IQR to mean/SD
- when in doubt, compute and compare
For example, which summary statistics are best to use below?
STAT218
Descriptive statistics Quantitative and graphical techniques for summarizing data