Statistiek I

Descriptive statistics

Martijn Wieling
University of Groningen

Question 1: last lecture

Last lecture

  • Why use statistics?
    • Summarize data (descriptive statistics)
    • Assess relationships in data (inferential statistics)
  • Introduction to RStudio and R
    • Variables and functions
    • Import, view and modify data in R
    • Some visualization and statistics in R

This lecture

  • Descriptive statistics vs. inferential statistics
  • Sample vs. population
  • (Types of) variables
  • Distribution of a variable
  • Measures of central tendency
  • Measures of variation
  • Standardized scores
  • Checking for a normal distribution

Descriptive vs. inferential statistics

  • Descriptive statistics:
    • Statistics used to describe (sample) data without further conclusions
      • Measures of central tendency: Mean, median, mode
      • Measures of variation (or spread): range, IQR, variance, standard deviation
  • Inferential statistics:
    • Describe data of sample in order to infer patterns in the population
      • Statistical tests: \(t\)-test, \(\chi^2\)-test, etc.

Question 2

Sample vs. population

Why study a sample?

  • Studying the whole population is (almost always) practically impossible
  • Sample is a (selected) subset of population and thus more accessible
    • Selection of representative sample is very important!

Question 3

Variables and values

  • Statistics always involve variables
    • Relations: involve two variables (e.g., English grade vs. English score)
  • The values of the variables indicate properties of the cases (i.e. the individuals or entities you study)
Variable Example values
Gender male, female
English grade 4.6, 5.5, 6.3, 7.2, ...
Year of birth 1990, 1991, 1993, ...
Native language Dutch, German, English, ...

Tabular representation of data

  • Each case is shown in a row
  • Each variable in a column
  • For part of our data:
participant year gender bl_edu study english_grade english_score
1 1 2017 F N LING 6 4.31
2 2 2017 M N LING 7 6.42
3 3 2017 M N LING 8 8.17
4 4 2017 F N CIS 7 6.99
5 5 2017 F N LING 7 6.12
6 6 2017 F N LING 8 7.35

Types of variables: nominal and ordinal

  • Nominal (categorical) scale: unordered categories
    • Gender (binary: only two categories), Native language, etc.
  • Ordinal: ordered (ranked) scale, but amount of difference unclear
    • Rank of English profiency (in class), Likert scale (Rate on a scale from 1 to 5...)

Types of variables: numerical

  • Interval scale: numerical with meaningful difference but no true 0
    • Year of birth, temperature in Celsius
  • Ratio scale: numerical with meaningful difference and true 0
    • Number of questions correct, age
  • Scale of variable determines possible statistics
    • E.g., mean age is possible, but not mean native language

Question 4

Types of variables: summary

Question 5

Characterizing nom. variables: (relative) freq. table

table(dat$gender)  # absolute frequencies
# 
#   F   M 
# 222  93
prop.table(table(dat$gender))  # relative frequencies
# 
#       F       M 
# 0.70476 0.29524

Characterizing nominal variables: visualization

par(mfrow = c(1, 2))
barplot(table(dat$gender), col = c("pink", "lightblue"), main = "abs.frq.")
barplot(prop.table(table(dat$gender)), col = c("pink", "lightblue"), main = "rel.frq.")

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Bad practice: pie charts

pie(table(dat$study), col = c("red", "green", "blue", "yellow"))

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Question 6

Characterizing numerical variables: distribution

  • We are generally not interested in individual values of a variable, but rather all values and their frequency
  • This is captured by a distribution
    • Famous distribution: Normal distribution ("bell-shaped" curve): e.g., IQ scores

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Distribution

  • Distribution of a variable shows variability of variable (i.e. frequency of values)
table(dat$english_grade)
# 
#   5 5.5   6 6.5   7 7.5   8 8.5   9 9.5 
#   3   1  48   6 126  10  96   7  17   1

Visualizing a distribution: histogram

  • Histogram shows frequency of all values in groups: \((a,b]\)
    • Look for general pattern, symmetry, outliers
hist(dat$english_grade, xlab = "English grade", main = "")

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Visualizing a distribution: density curve

  • Density curve shows area proportional to the relative frequency
plot(density(dat$english_grade), main = "", xlab = "English grade")

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Interpreting a density curve

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  • The total area under a density curve is equal to 1
  • A density curve does not provide information about the frequency of one value
    • E.g., there might be no one who has scored a grade of exactly 6.1
  • It only provides information about an interval
    • E.g., more than 50% of the grades are between 5.5 and 7.5

Interpreting a density curve: normal distribution

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  • The normal distribution has convenient characteristics
    • Completely symmetric
    • Red area: (about) 68%
    • Red and green area: (about) 95%

Characterizing the distribution of numerical variables

  • A distribution can also be characterized by measures of center and variation
    • (skewness measures the symmetry of the distribution; not covered in this course)

Characterizing numerical variables: central tendency

  • Mode: most frequent element (for nominal data: only meaningful measure)
  • Median: when data is sorted from small to large, it is the middle value
  • Mean: arithmetical average

\[\bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n} = \frac{1}{n}\sum\limits_{i=1}^n x_i\]

  • (You need to be able to calculate the mean of a series of values by hand, so it is useful to remember the formula)

Question 7

Measures of center may have very different values

Central tendency in R

mean(dat$english_grade)  # arithmetic average
# [1] 7.2813
median(dat$english_grade)
# [1] 7
# no built-in function to get mode: new function
my_mode <- function(x) {
    counts <- table(x)
    names(which(counts == max(counts)))
}
my_mode(dat$english_grade)
# [1] "7"

Measure of variation: quantiles

  • Quantiles: cutpoints to divide the sorted data in subsets of equal size
    • Quartiles: three cutpoints to divide the data in four equal-sized sets
      • \(q_1\) (1st quartile): cutpoint between 1st and 2nd group
      • \(q_2\) (2nd quartile): cutpoint between 2nd and 3rd group (= median!)
      • \(q_3\) (3rd quartile): cutpoint between 3rd and 4th group
    • Percentiles: divide data in hundred equal-sized subsets
      • \(q_1\) = 25th percentile
      • \(q_2\) (= median) = 50th percentile
      • Score at $n$th percentile is better than \(n\)% of scores
quantile(dat$english_grade)  # default: quartiles
#   0%  25%  50%  75% 100% 
#  5.0  7.0  7.0  8.0  9.5

Measure of variation: range

  • Minimum, maximum: lowest and highest value
  • Range: difference between minimum and maximum
  • Interquartile range (IQR): \(q_3\) - \(q_1\)

Visualizing variation: box plot (box-and-whisker plot)

  • A box plot is used to visualize variation of a variable
    • Box (IQR): \(q_1\) (bottom), median (thickest line), \(q_3\) (top)
      • (In example below, \(q_1\) and median have the same value)
    • Whiskers: maximum (top) and minimum (bottom) non-outlier value
    • Circle(s): outliers (> 1.5 IQR distance from box)
boxplot(dat$english_grade, col = "red")

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Important measure of variation: variance

  • Deviation: difference between mean and individual value
  • Variance: average squared deviation
    • Squared in order to make negative differences positive
    • Population variance (with \(\mu\) = population mean): \[\sigma^2 = \frac{1}{n}\sum\limits_{i=1}^n (x_i - \mu)^2\]
    • As sample mean (\(\bar{x}\)) is approximation of population mean (\(\mu\)), sample variance formula contains division by \(n-1\) (results in slightly higher variance): \[s^2 = \frac{1}{n-1}\sum\limits_{i=1}^n (x_i - \bar{x})^2\]

Important measure of variation: standard deviation

  • Standard deviation is square root of variance \[\sigma = \sqrt{\sigma^2} = \sqrt{\frac{1}{n}\sum\limits_{i=1}^n (x_i - \mu)^2}\] \[s = \sqrt{s^2} = \sqrt{\frac{1}{n-1}\sum\limits_{i=1}^n (x_i - \bar{x})^2}\]

Question 8

Measures of variation (or spread) in R (1)

min(dat$english_grade)  # minimum value
# [1] 5
max(dat$english_grade)  # maximum value
# [1] 9.5
range(dat$english_grade)  # returns minimum and maximum value
# [1] 5.0 9.5
diff(range(dat$english_grade))  # returns difference between min. and max.
# [1] 4.5

Measures of variation (or spread) in R (2)

IQR(dat$english_grade)  # interquartile range
# [1] 1
var(dat$english_grade)  # sample variance
# [1] 0.71962
sd(dat$english_grade)  # sample standard deviation
# [1] 0.8483
sd(dat$english_grade) == sqrt(var(dat$english_grade))  # std. dev. = sqrt of var.?
# [1] TRUE

Normal distribution and standard deviation (1)

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\(P(\mu - \sigma \leq x \leq \mu + \sigma) \approx 68\%\)     (34 + 34)
\(P(\mu - 2\sigma \leq x \leq \mu + 2\sigma) \approx 95\%\)     (34 + 34 + 13.5 + 13.5)
\(P(\mu - 3\sigma \leq x \leq \mu + 3\sigma) \approx 99.7\%\)     (34 + 34 + 13.5 + 13.5 + 2.35 + 2.35)

  • It is important to remember these characteristics of the normal distribution!

Normal distribution and standard deviation (2)

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\(P(85 \leq \rm{IQ} \leq 115) \approx 68\%\)     (34 + 34)
\(P(70 \leq \rm{IQ} \leq 130) \approx 95\%\)     (34 + 34 + 13.5 + 13.5)
\(P(55 \leq \rm{IQ} \leq 145) \approx 99.7\%\)     (34 + 34 + 13.5 + 13.5 + 2.35 + 2.35)

  • IQ scores are normally distributed with mean 100 and standard deviation 15

Standardized scores

  • Standardization helps facilitate interpretation
  • E.g., how to interpret: "Emma's score is 112" and "Tom's score is 105"
  • Interpretation should be done with respect to mean \(\mu\) and standard deviation \(\sigma\)
    • Raw scores can be transformed to standardized scores (\(z\)-scores or \(z\)-values) \[z = \frac{x - \mu}{\sigma} = \frac{\rm{deviation}}{\rm{standard}\,\rm{deviation}}\]
    • Interpretation: difference of value from mean in number of standard deviations
  • (Note that you have to be able to calculate \(z\)-scores!)

Calculating standardized values

  • Suppose \(\mu = 108\), \(\sigma = 4\), then: \[z_{112} = \frac{x - \mu}{\sigma} = \frac{112 - 108}{4} = 1\] \[z_{105} = \frac{105 - 108}{4} = -0.75\]
  • \(z\) shows distance from mean in number of standard deviations

Question 9

Distribution of standardized variables

  • If we transform all raw scores of a variable into \(z\)-scores using: \[z = \frac{x - \mu}{\sigma} = \frac{\rm{deviation}}{\rm{standard}\,\rm{deviation}}\]
  • We obtain a new transformed variable whose
    • Mean is 0
    • Standard deviation is 1
  • In sum: \(z\)-score = distance from \(\mu\) in \(\sigma\)'s
  • \(z\)-scores are useful for interpretation and hypothesis testing (next lecture)

Standardizing a variable in R

dat$english_grade.z = scale(dat$english_grade)  # scale: calculates z-scores
mean(dat$english_grade.z)  # should be 0
# [1] 0
sd(dat$english_grade.z)  # should be 1
# [1] 1

Standard normal distribution

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\(P(-1 \leq z \leq 1) \approx 68\%\)     (34 + 34)
\(P(-2 \leq z \leq 2) \approx 95\%\)     (34 + 34 + 13.5 + 13.5)
\(P(-3 \leq z \leq 3) \approx 99.7\%\)     (34 + 34 + 13.5 + 13.5 + 2.35 + 2.35)

For comparison: normal distribution

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\(P(\mu - \sigma \leq x \leq \mu + \sigma) \approx 68\%\)     (34 + 34)
\(P(\mu - 2\sigma \leq x \leq \mu + 2\sigma) \approx 95\%\)     (34 + 34 + 13.5 + 13.5)
\(P(\mu - 3\sigma \leq x \leq \mu + 3\sigma) \approx 99.7\%\)     (34 + 34 + 13.5 + 13.5 + 2.35 + 2.35)

Percentiles and \(z\)-scores

  • If distribution is normal then \(z\)-scores correspond to percentiles
  • The function qnorm returns the \(z\)-values for a certain proportion (percentile / 100)
qnorm(95/100)  # z-value associated with 95th percentile
# [1] 1.6449

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Question 10

\(z\)-scores and percentiles

  • The function pnorm returns the proportion of data < a specified \(z\)-value
    • the percentile can be found by multiplying with 100
100 * pnorm(1.6449)
# [1] 95

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Some calculations with \(z\)-scores

  • What proportion of values have a \(z\)-value of at least 1.64?
    • \(P(z \geq 1.64)\)
1 - pnorm(1.64)
# [1] 0.050503
  • What proportion of values are located between \(z\)-values between -2 and 2?
    • \(P(-2 \leq z < 2)\)
pnorm(2) - pnorm(-2)
# [1] 0.9545

Visualization

pnorm(2)
# [1] 0.97725
pnorm(-2)
# [1] 0.02275

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Question 11

Checking normality assumption

  • Some statistical tests (e.g., \(t\)-test) require that the data is (roughly) normally distributed
  • How to test this?
    • Using visual inspection of a normal quantile plot (or: quantile-quantile plot)
      • A straight line in this graph indicates a (roughly) normal distribution
    • Using the Shapiro-Wilk test (covered in lecture 5)

Normal quantile plot: how it works

  • Sort the data from smallest to largest to determine quantiles (e.g., percentiles)
    • E.g., median for 50th percentile
  • Calculate \(z\)-values belonging to the quantiles (e.g., percentiles) of a standard normal distribution
    • E.g., \(z =\) 0 for 50th percentile, \(z =\) 2 for 97.5th percentile, etc.
  • Plot data values (\(y\)-axis) against normal quantile values (\(x\)-axis)
    • If points on (or close to) straight line: values normally distributed
  • (Note: you need to be able to interpret a quantile-quantile plot, but you don't have to be able to construct the plot manually)

Normal quantile plot in R: English grades

qqnorm(dat$english_grade)
qqline(dat$english_grade)

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  • Distribution not normal for the English grades

Normal quantile plot in R: English scores

qqnorm(dat$english_score)
qqline(dat$english_score)

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  • Distribution roughly normal for the English scores

Recap

  • In this lecture, we've covered
    • Descriptive statistics vs. inferential statistics
    • Sample vs. population
    • Four types of variables
    • Distribution of a variable
    • Measures of central tendency
    • Measures of variation
    • Standardized scores
    • How to check for a normal distribution
  • In the lab session, you will experiment with descriptive statistics
  • Next lecture: Sampling

Please evaluate this lecture!

Exam question

Questions?

Thank you for your attention!

http://www.martijnwieling.nl
m.b.wieling@rug.nl