In exploring statistical distributions, we focus on the following:

  • what influences the shape of a distribution
  • calculate probability from a given quantile
  • calculate quantiles out of given probability

To explore the above 3 concepts, we have defined functions for the following distributions:

  • Normal
  • Binomial
  • Chi Square
  • F
  • t

Normal Distribution

Distribution Shape

Visualize how changes in mean and standard deviation affect the shape of the normal distribution.

Input
  • mean: mean of the normal distribution
  • sd: standard deviation of the normal distribution
Output
  • Normal distribution plot
dist_norm_plot(mean = 2, sd = 0.6)

Percentiles

Calculate and visualize quantiles out of given probability.

Input
  • probs: a probability value
  • mean: mean of the normal distribution
  • sd: standard deviation of the normal distribution
  • type: lower/upper tail

Suppose X, the grade on a exam, is normally distributed with mean 60 and standard deviation 3. The teacher wants to give 10% of the class an A. What should be the cutoff to determine who gets an A?

dist_norm_perc(0.10, 60, 3, 'upper')

The teacher wants to give lower 15% of the class a D. What cutoff should the teacher use to determine who gets an D?

dist_norm_perc(0.85, 60, 3, 'lower')

The teacher wants to give middle 50% of the class a B. What cutoff should the teacher use to determine who gets an B?

dist_norm_perc(0.5, 60, 3, 'both')

Probabilities

Calculate and visualize probability from a given quantile

Input
  • perc: a quantile value
  • mean: mean of the normal distribution
  • sd: standard deviation of the normal distribution
  • type: lower/upper/both tail

Let X be the IQ of a randomly selected student of a school. Assume X ~ N(90, 4). What is the probability that a randomly selected student has an IQ below 80?

dist_norm_prob(80, mean = 90, sd = 4)

What is the probability that a randomly selected student has an IQ above 100?

dist_norm_prob(100, mean = 90, sd = 4, type = 'upper')

What is the probability that a randomly selected student has an IQ between 85 and 100?

dist_norm_prob(c(85, 100), mean = 90, sd = 4, type = 'both')

Binomial Distribution

Distribution Shape

Visualize how changes in number of trials and the probability of success affect the shape of the binomial distribution.

Percentiles

Calculate and visualize quantiles out of given probability

Input
  • p: a single aggregated probability of multiple trials
  • n: the number of trials
  • tp: the probability of success in a trial
  • type: lower/upper tail
dist_binom_perc(10, 0.5, 0.05)

dist_binom_perc(10, 0.5, 0.05, 'upper')

Probabilities

Calculate and visualize probability from a given quantile

Input
  • p: probability of success
  • n: the number of trials
  • s: number of success in a trial
  • type: lower/upper/interval/exact tail

Assume twenty-percent (20%) of Magemill have no health insurance. Randomly sample n = 12 Magemillians. Let X denote the number in the sample with no health insurance. What is the probability that exactly 4 of the 15 sampled have no health insurance?

dist_binom_prob(12, 0.2, 4, type = 'exact')

What is the probability that at most one of those sampled has no health insurance?

dist_binom_prob(12, 0.2, 1, 'lower')

What is the probability that more than seven have no health insurance?

dist_binom_prob(12, 0.2, 8, 'upper')

What is the probability that fewer than 5 have no health insurance?

dist_binom_prob(12, 0.2, c(0, 4), 'interval')

Chi Square Distribution

Distribution Shape

Visualize how changes in degrees of freedom affect the shape of the chi square distribution.

dist_chi_plot(df = 5, normal = TRUE)

Percentiles

Calculate quantiles out of given probability

Input
  • probs: a probability value
  • df: degrees of freedom
  • type: lower/upper tail

Let X be a chi-square random variable with 8 degrees of freedom. What is the upper fifth percentile?

dist_chi_perc(0.05, 8, 'upper')

What is the tenth percentile?

dist_chi_perc(0.10, 8, 'lower')

Probability

Calculate probability from a given quantile.

Input
  • perc: a quantile value
  • df: degrees of freedom
  • type: lower/upper tail

What is the probability that a chi-square random variable with 12 degrees of freedom is greater than 8.79?

dist_chi_prob(8.79, 12, 'upper')

What is the probability that a chi-square random variable with 12 degrees of freedom is greater than 8.62?

dist_chi_prob(8.62, 12, 'lower')

F Distribution

Distribution Shape

Visualize how changes in degrees of freedom affect the shape of the F distribution.

dist_f_plot(6, 10, normal = TRUE)

Percentiles

Calculate quantiles out of given probability

Input
  • probs: a probability value
  • num_df: nmerator degrees of freedom
  • den_df: denominator degrees of freedom
  • type: lower/upper tail

Let X be an F random variable with 4 numerator degrees of freedom and 5 denominator degrees of freedom. What is the upper twenth percentile?

dist_f_perc(0.20, 4, 5, 'upper')

What is the 35th percentile?

dist_f_perc(0.35, 4, 5, 'lower')

Probabilities

Calculate probability from a given quantile.

Input
  • perc: a quantile value
  • num_df: nmerator degrees of freedom
  • den_df: denominator degrees of freedom
  • type: lower/upper tail

What is the probability that an F random variable with 4 numerator degrees of freedom and 5 denominator degrees of freedom is greater than 3.89?

dist_f_prob(3.89, 4, 5, 'upper')

What is the probability that an F random variable with 4 numerator degrees of freedom and 5 denominator degrees of freedom is less than 2.63?

dist_f_prob(2.63, 4, 5, 'lower')

t Distribution

Distribution Shape

Visualize how degrees of freedom affect the shape of t distribution.

dist_t_plot(df = 8)

Percentiles

Calculate quantiles out of given probability

Input
  • probs: a probability value
  • df: degrees of freedom
  • type: lower/upper/both tail

What is the upper fifteenth percentile?

dist_t_perc(0.15, 8, 'upper')

What is the eleventh percentile?

dist_t_perc(0.11, 8, 'lower')

What is the area of the curve that has 95% of the t values?

dist_t_perc(0.8, 8, 'both')

Probabilities

Calculate probability from a given quantile.

Input
  • perc: a quantile value
  • df: degrees of freedom
  • type: lower/upper/interval/both tail

Let T follow a t-distribution with r = 6 df.

What is the probability that the value of T is less than 2?

dist_t_prob(2, 6, 'lower')

What is the probability that the value of T is greater than 2?

dist_t_prob(2, 6, 'upper')

What is the probability that the value of T is between -2 and 2?

dist_t_prob(2, 6, 'both')

What is the probability that the absolute value of T is greater than 2?

dist_t_prob(2, 6, 'interval')