Statistics

• Statistics: The science of gathering, presenting, analysing and interpreting data
• Some basic terms:

  – Population – the whole

  – Census – data gathered from the entire population

  – Sample – a subset of the population
• Descriptive statistics – summarizes the characteristics of a dataset
• Inferential statistics – using sample data to reach conclusions about the population from which the sample was drawn.
• Parameter

  – Descriptive measure of the population

  – Represented by Greek letters

  e.g.

  μ - population mean

  σ² - population variance

  σ - population standard deviation
• Statistic

  – Descriptive measure of a sample

  – Represented by roman letters

  e.g.

  x̄ - sample means

  s² - sample variance

  s - sample standard deviation
• Elements– the entities on which data are collected
• Observations– the set of measurements obtained for a particular element
• Variable– a characteristic of interest for the element
• Levels of data - Nominal

  • Labels or names used to identify an attribute of the element.
  • Data cannot be ordered
• Levels of data - Ordinal

  • Numbers/categories are used to indicate a rank or order.
  • Relative magnitudes are meaningful
• Levels of data - Interval

  • Data shows the properties of ordinal data
  • Interval between the values is expressed in terms of a fixed unit of measure
  • Always numeric
• Levels of data - Ratio

  • Has properties of interval data
  • The location of origin is zero – nothing exists for the variable at zero point
  • Variables such height, time and weight
• Different types of dataset –Time series

  Values of a variable at different points in time
• Different types of dataset –Panel data

  Observations on individuals overtime
• Bar chart: the value of each category in categorical data is represented by a bar
• Pie chart: the area of each section represents the relative frequency
• Graphical method - histogram

  • Shows data summarised in frequency density (frequency/class width)
  • It provides information about the shape of a distribution
  • Similar to a bar chart, but it corrects for differences in class widths. If all class widths are identical, a bar chart and histogram have the same shape
• Ogive

  Cumulative distribution plot - distribution curve in which the frequencies are cumulative
• Group data into intervals:

  • Frequency: Number of times a particular value occurs
  • Cumulative frequency: the sum of successive frequencies
  • Relative frequency: Proportion of observations in each class
  • Cumulative relative frequency: The sum of successive relative frequencies
• Arithmetic mean – population mean

  Can be applied to interval and ratio data

  Is affected by all the values in the dataset (potential problem: extreme values/outliers in the dataset)

  For a population:

  𝜇 = ∑𝑥𝑖 / N

  Where:

  𝜇 - population mean

  𝑥𝑖 - value of the variable

  𝑁 - number of observations
• Arithmetic mean – sample mean

  For a sample :

  x̄ = (𝑥1+𝑥2 + 𝑥3 + ⋯ 𝑥𝑛)/𝑛

  Or

  x̄ = ∑ 𝑥𝑖 / n

  Where

  x̄ - sample mean

  𝑥𝑖 - value of the variable

  n - number of observations
• Arithmetic mean – grouped data

  Grouped data: values have been organised into a frequency distribution

  For a population: 𝜇 = ∑ fi𝑥mid_i / N

  For a sample: x̄ = ∑ fi𝑥mid_i / n

  Where

  fi : frequency in each class i

  xmid_i : the mid-point value of each class i

  n: number of observations
• Median

  • The value that divides the ordered sample into two parts, with equal numbers of observations in each part
  • Can be applied to ordinal, interval and ratio data
  • Not affected by extreme values
• Median – cont’d

  The median’s position in an ordered row:

  Sample with an odd number of observations, the (𝑛𝑛 + 1)/2th observation:

  12, 13, 13, 16, 17, 19, 20(7 + 1)/2 = 4th observation

  Sample with an even number of observations, the average value between the(𝑛/2)th and (𝑛/2 + 1)th ordered observation:

  12, 13, 13, 16, 17, 19, 20, 28

  Median is between 16 and 17: 16.5
• Median – grouped data

  2 steps involved:

  Calculate the class interval that contains the median observation

  2) Calculate the value of median using the following formula:

  median = xl + (xu - xl) {((N+1)/2) - F / f}

  Where:

  xl - lower limit of class interval

  xu - upper limit of class interval

  N - no. of observations

  F - cumulative frequency of the class intervals up to the one containing the median

  f - the frequency for the class interval containing the median
• Mode

  The most frequently occurring value in the dataset

  Can be applied to all levels of data (Nominal, Ordinal,Interval and Ratio)

  Bi-modal: two values have the highest number of occurrences in the dataset

  Multi-modal: three or more values have the highest number of occurrences in the dataset
• Additional measures – quartiles

  Quartiles divide a distribution into four equal parts

  Additional measures – quartiles

  Quartiles divide a distribution into four equal parts

  Procedure:

  1. Calculate the class interval which contains the quartile
  2. Calculate the quartiles using the following formulae:

  Q1 = xl + (xu - xl) {((N + 1) / 4) - F / f}

  Q3 = xl + (xu - xl) {((N + 1)3 / 4) - F / f}
• Choice of measures of central location

  • Depends on the purpose/question being asked

  • Arithmetic mean is used the most widely

  • If observations are symmetrically distributed (and unimodal), mean, median and mode are identical

  • If data is not symmetrically distributed, then “skewed”distribution
• Range:

  •The difference between the largest and smallest observation

  •Simplest, easy to calculate

  •Ignores all data points except the extremes
• Interquartile Range:

  •The difference between 1st and 3rd quartile

  • Defines the range of the middle 50% of observations
• Mean Absolute Deviation (MAD)

  The average of the absolute deviations from the mean

  Focus on the dispersion around the central location of the data

  For a population: MAD = ∑ | x - 𝜇 | / N

  For a sample: MAD = ∑ | x - x̄ | / n
• Note: In Business/Investment Evaluation....

  Mean considered a measure of return

  Standard Deviation considered a measure of risk, asit measures the range of an investment's performance.

  The greater the standard deviation, the greater the investment's volatility.
• Variance for a population

  A measure that makes use of all of the information available

  The population variance is given as:

  𝜎2 = ∑ (x - 𝜇)^2 / N

  or

  for grouped data:

  𝜎2 = ∑ f (xmid - ��)^2 / N

  Where:

  𝜎2 - variance of the population

  x - each value in the dataset

  xmid - mid-point value of each class interval for grouped data

  𝜇 - population mean

  N - number of observations

  f - frequency in each class interval for grouped data
• Standard deviation for a population

  The square root of variance.

  Population standard deviation is given as:

  𝜎 = square root of ∑ (x - 𝜇)^2 / N

  for grouped data:

  𝜎 = square root of ∑ f(xmid - 𝜇)^2 / N

  Where:

  𝜎 - standard deviation of the population

  x - each value in the dataset

  xmid - mid-point value of each class interval for grouped data

  𝜇 - population mean

  N - number of observations

  f - frequency in each class interval for grouped data
• Standard deviation for a sample

  Sample standard deviation is given as:

  s = square root of ∑ ( x - x̄)^2 / n - 1

  or

  for grouped data:

  s = square root of ∑ f( xmid- x̄)^2 / n - 1

  Where:

  s - standard deviation of the sample

  x - each value in the dataset;

  xmid - mid-point value of each class interval for grouped data

  x̄ - sample mean

  n - number of observations

  f - frequency in each class interval for grouped data
• Variance and standard deviation –simplified formulae

  Alternatively, the following simplified formulae can be used(recommended):

  Variance of a population:

  𝜎^2 = ∑ x^2 - N𝜇^2 / N

  or for grouped data:

  𝜎^2 = ∑ fx^2mid - N𝜇^2 / N

  Variance of a sample:

  s^2 = ∑ x^2 - nx̄^2 / n

  or for grouped data:

  s^2 = ∑f x^2mid - nx̄^2 / n

  Standard deviation is the square root of variance
• Coefficient of variation

  A measure of relative dispersion (independent of units of measurements):

  For a population:

  cofficient of variation = 𝜎/𝜇

  For a sample:

  cofficient of variation = s/x̄

  Where:

  𝜎 - standard deviation of the population

  𝜇 - population mean

  s - standard deviation of the sample

  x̄ - sample mean
• • A numerical measure of the likelihood that an event will occur

  P(E) = No of favourable occurrences / Total number of possible outcomes

  • Probability values are assigned on a scale from 0 to 1
• The frequentist view

  The proportion of an outcome obtained in the trials as the number of trials approaches infinity Again, using the previous example: what is the probability of “heads” occurring on the toss of coin?