Week 1: Exploring Data

Introduction to Statistics

• Statistics: Quantifying the world in an objective way for good decision-making.

• Modules:

  • Descriptive Statistics and data communication (Module 1).

  • Inferential Statistics (Module 2).

  • Hypothesis Testing (Module 3).

  • Regression Analysis (Module 4).

  • Data ethics (Module 5).

Week 1: Exploring Data - Foundations of Descriptive Statistics

• Data Types and Visualization

Preliminary Terms and Definitions

• Variable: A characteristic, number, or quantity that can be measured or counted.

• Random variable: A variable whose outcome is unknown before data collection

  • Example: income of an Australian household.

• Population: The complete pool of a particular random variable

  • Example: income of all Australian households.

• Sample: A subset of the population

  • Example: income of 100 households.

• Goal: Describe and visualize information contained in different types of variables.

Types of Data

• Variables are broadly classified as qualitative/categorical or quantitative/numerical.

• Qualitative/Categorical:

  • Nominal: Categories with no natural ordering

    • Example: 0/1 variable for male/female.

  • Ordinal: Categories with a natural order, but numbers are meaningless

    • Example: Agreement scale (don’t agree=-1/somewhat agree=0/completely agree=1).

• Quantitative/Numerical:

  • Discrete: Values can be listed (not infinitely divisible), often from counting

    • Example: Number of children in a household (0, 1, 2, 3, …).

  • Continuous: Can take an infinite number of values within a range, often from measurement.

    • Example: Heights of professional basketball players.

Frequency Distributions

• For qualitative/categorical data, visualize via a table displaying frequencies.

• Example Table:

  • Material Status of home loan applicants:

    • Single: Frequency 102, Relative Frequency 0.1262, Percent Frequency 12.62

    • Married: Frequency 341, Relative Frequency 0.4220, Percent Frequency 42.20

    • Widowed: Frequency 155, Relative Frequency 0.1918, Percent Frequency 19.18

    • De Facto: Frequency 50, Relative Frequency 0.0619, Percent Frequency 6.19

    • Separated: Frequency 40, Relative Frequency 0.0495, Percent Frequency 4.95

    • Divorced: Frequency 120, Relative Frequency 0.1485, Percent Frequency 14.85

    • Total: Frequency 808, Relative Frequency 1, Percent Frequency 100

Key Terms

• Frequency counts: Total occurrences for each category.

• Relative frequency: Fraction/proportion of total data items in a category.

• Percent frequency: Relative frequency × 100 (%).

Excel Function

• Use COUNTIF(range, values) to obtain frequency counts

• Example Formula: =COUNTIF(I$10:I$389, $D2)

Data Visualization: Histograms

• Commonly used for continuous variables.

• Steps:

  1. Choose a bandwidth/bin size to group incomes into equally spaced categories

     • e.g., $0-100, $101-$200, $201-$300 etc.

  2. Plot frequencies for each group in a bar chart

     • Frequencies on the y-axis, categories on the x-axis.

Histogram Example

• Income brackets and Frequencies:

  • (6, 526]: Frequency near 35

  • (526, 1046]: Frequency around 25

  • (1046, 1566]: Frequency around 15

  • (1566, 2086]: Frequency around 10

  • (2086, 2606]: Frequency around 5

  • (2606, 3126]: Frequency around 2

Histogram Considerations

• In histograms for numerical data, ensure that the income ranges are contiguous.

• Gaps or overlaps in income ranges are not accurate representation of the data

Data Visualization: Bar Chart

• Visually similar to a histogram, but:

  • Categories need not be equally ranged continuous values.

  • The y-axis can represent things other than frequency.

  • Usually whitespace between the bars.

• Example: Marital Status of home loan applicants:

  • Single: ~100

  • Married: ~350

  • Widowed: ~150

  • De Facto: ~50

  • Separated: ~40

  • Divorced: ~120

Data Visualization: Pie Chart

• Way to visualize categorical data, frequencies shown as segments of a circle

• Example: Marital Status of home loan applicants (same categories as bar chart)

• Tip: Pie charts are rarely a good idea, and never when there are a large number of categories

Summary Statistics: Central Tendency

• Describing Data: Central Tendency, Variability, Skewness

Notation

• Random variables: Denoted by capital letters (𝑋, 𝑌).

  • 𝑋: Number of children in a household.

  • 𝑌: Amount of time spent by the husband on housework per day.

• Realizations/observations of a random variable: Lowercase letters with subscript (𝑥𝑖, 𝑦𝑖).

  • 𝑥1: Number of children in household 1.

  • 𝑦137: Amount of time spent by husband 137 on housework per day.

• 𝑁 and 𝑛: Denote the size or number of observations.

  • 𝑁: Population size (usually very large, can be infinite).

  • 𝑛: Sample size, i.e., the number of data points collected in a sample.

Central Tendency

• Definition: Measures of central tendency provide information about the center of the distribution of a random variable; indicate a typical, middle, or average value. (Measures of location)

  1. Mean: Arithmetic average value.

  2. Mode: Most commonly occurring value.

  3. Median: Middle value in an ordered array.

Central Tendency: Mean

• Population Mean: Denoted by $μ$ or $E(X)$, the expectation of $X$. Computed by:

  • $μ = E(X) = \frac{x<em>1 + x</em>2 + \cdots + x<em>N}{N} = \frac{1}{N}\sum</em>{i=1}^{N} x_i$

• Sample Mean: Denoted by $\bar{X}$, called X bar. Computed by:

  • $\bar{X} = \frac{x<em>1 + x</em>2 + \cdots + x<em>n}{n} = \frac{1}{n}\sum</em>{i=1}^{n} x_i$

Example

• Random variable: Height of females aged between 25 and 40.

• John has a sample of randomly chosen females aged 25 and 40:

  • Heights are 157cm, 163cm, 166cm, 148cm, 174cm, 165cm, 168cm.

• Sample size $n = 7$. ($x<em>1 = 157\text{cm}, \dots, x</em>7 = 168\text{cm}$)

• Sample mean: $\bar{X} = \frac{157 + 163 + 166 + 148 + 174 + 165 + 168}{7} = 163 \text{cm}$

Example: Gamble

• Gamble: Tossing a fair coin.

  • Heads: Receive $10.

  • Tails: Pay $10.

• Scenario: Play the gamble 100 times; 60 heads, 40 tails.

• Sample mean: $\bar{X} = \frac{60 \times 10 + 40 \times (-10)}{100} = \frac{600 - 400}{100} = 2$

Example: Population Mean

• Consider the same gamble: fair coin toss, receive/pay $10 for heads/tails

• Population mean:

  • $E(X) = 0.5 \times 10 + 0.5 \times -10 = 0$

Central Tendency: Mode

• The mode is the most commonly occurring value.

• Example: Waiting times of people in a queue (minutes):

  • 2, 3, 3, 3, 4, 2, 2, 2, 2, 3, 3, 3, 1, 1 (ordered: 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4)

• What is the mode? 3 (occurs six times).

• Random variables with two modes are bimodal; with multiple modes, multimodal.

Central Tendency: Median

• The median is the middle value in an ordered array.

• Example: Waiting times (minutes):

  • 2, 3, 3, 3, 4, 2, 2, 2, 2, 3, 3, 3, 1, 1 (ordered: 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4)

• What is the median? Median is 2.5 minutes (lies in the middle of the 14 numbers).

  • Median = $\frac{2+3}{2} = 2.5$

Central Tendency: Qualitative Data

• Example: University major of employees (1=marketing, 2=finance, 3=economics, 4=law, 5=others).

• Recorded data: 2, 5, 3, 1, 4, 2, 5, 3, 4, 2, 1 (ordered: 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 5).

• Mode is 2, median is 3, mean is 2.9091.

• Question: Which measure of central tendency is the most appropriate?

Summary Statistics: Variability

• Describing Data: Central Tendency, Variability, Skewness

Descriptive Statistic: Variability

• Definition: Measures of variability provide information about how dispersed the values of a random variable are around the mean (measures of scale, spread, dispersion, or risk).

  1. Variance (Var): Average of squared distance from the mean.

  2. Standard deviation (std): Square root of variance.

  3. Coefficient of variation: $\frac{\text{std}}{\text{mean}} \times 100\%$. Measures risk per unit of expected return.

Variability: Example

• Question: Which stock to invest based on the data of their weekly returns?

• Stock X and Stock Y have $E(X) = E(Y) = 1.5\%$, meaning that every week both stocks are expected to grow 1.5% (on average). But which one do you prefer?

Variability: Formulas

• Population Variance: Denoted by $\sigma^2$ or $Var(X)$. Computed by:

  • $\sigma^2 = Var(X) = \frac{(x<em>1 - \mu)^2 + \cdots + (x</em>N - \mu)^2}{N} = \frac{1}{N}\sum<em>{i=1}^{N} (x</em>i - \mu)^2$

• Sample Variance: Denoted by $s^2$. Computed by:

  • $s^2 = \frac{(x<em>1 - \bar{X})^2 + \cdots + (x</em>n - \bar{X})^2}{n - 1} = \frac{1}{n - 1}\sum<em>{i=1}^{n} (x</em>i - \bar{X})^2$

Variability: Variance

• Computes the average squared distance between data points and their mean.

• Given: data points +2, +4, -6 and $\bar{X} = 12$

• $x<em>1 = 14, x</em>1 - \bar{X} = 2, (x_1 - \bar{X})^2 = 4$

• $x<em>3 = 16, x</em>3 - \bar{X} = 4, (x_3 - \bar{X})^2 = 16$

• $x<em>2 = 6, x</em>2 - \bar{X} = -6, (x_2 - \bar{X})^2 = 36$

Example: Waiting Time

• $X$: waiting time of people in a queue (in minutes)

• Observations: 12, 9, 8, 8, 11, 9, 10, 9, 14, 9, 9, 7, 10, 10, 14

• Population: $N = 15, \mu = \frac{1}{15}\sum<em>{i=1}^{15} x</em>i = 9.93 \text{ minutes}$. $\sigma^2 = \frac{1}{15}\sum<em>{i=1}^{15} (x</em>i - \mu)^2 = 3.929$

• Sample: $n = 15, \bar{X} = \frac{1}{15}\sum<em>{i=1}^{15} x</em>i = 9.93 \text{ minutes}$. $s^2 = \frac{1}{14}\sum<em>{i=1}^{15} (x</em>i - \bar{X})^2 = 4.210$

Variance: Remarks

• Q1: Why sum up or average out squared distance instead of distance?

  • Distance in different directions may cancel out, not suitable for measuring variability.

• Q2: What is the unit?

  • Distance such as $x_1 - \mu$ is in the unit of the data.

  • Squared distance such as $(x_1 - \mu)^2$ is in the unit of the data squared!

  • Example: Distance such as $x<em>1 - \mu = (12-9.93)$ is in minutes. Squared distance such as $(x</em>1 - \mu)^2 = (12-9.93)^2$ is in minutes squared.

Standard Deviation

• Population Standard Deviation: Denoted by $\sigma$ or $std(X)$. Computed by:

  • $\sigma = \sqrt{\sigma^2} = \sqrt{\frac{(x<em>1 - \mu)^2 + \cdots + (x</em>N - \mu)^2}{N}} = \sqrt{\frac{1}{N}\sum<em>{i=1}^{N} (x</em>i - \mu)^2}$

• Sample Standard Deviation: Denoted by $s$. Computed by:

  • $s = \sqrt{s^2} = \sqrt{\frac{(x<em>1 - \bar{X})^2 + \cdots + (x</em>n - \bar{X})^2}{n - 1}} = \sqrt{\frac{1}{n - 1}\sum<em>{i=1}^{n} (x</em>i - \bar{X})^2}$

Standard Deviation

• Standard deviation solves the problem of squared units.

• Has the same units as the original data.

• In the waiting example:

  • Population: $\sigma = \sqrt{3.929 \text{ minutes}^2} = 1.982 \text{ minutes}$

  • Sample: $s = \sqrt{4.210 \text{ minutes}^2} = 2.052 \text{ minutes}$

Standard Deviation - Example

• Variance and standard deviation measure how spread out the distribution of a random variable is.

• X: time spent on work, Y: time spent on leisure (per day) with 5 observations.

• Means are the same $(\bar{X} = \bar{Y} = 6)$. Variances are different (sX^2 = 2.5 < sY^2 = 12.5).

Coefficient of Variation

• Population CV(%): $CV = \frac{\sigma}{\mu} \times 100\%$. Sample CV(%): $CV = \frac{s}{\bar{X}} \times 100\%$.

• It is unit free because both the numerator and denominator have the same unit as the original data.

Coefficient of Variation - Example

• $X$: waiting time of people in a queue (in minutes)

• Observations: 12, 9, 8, 8, 11, 9, 10, 9, 14, 9, 9, 7, 10, 10, 14

• Population CV(%): $CV = \frac{1.982 \text{ minutes}}{9.93 \text{ minutes}} \times 100\% = 19.96\%$.

• Sample CV(%): $CV = \frac{2.052 \text{ minutes}}{9.93 \text{ minutes}} \times 100\% = 20.66\%$.

CV: Interpretation

• $CV$ is unit free. It measures standard deviation per unit of mean.

• Example: Time on leisure per day in hours vs in minutes (same coefficient of variation if regarding the data as a sample).

• Example: In finance when the random variable $X$ denotes asset returns, $CV$ measures risk per unit of expected return.

Variability: Excel

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Summary Statistics: Skewness

• Describing Data: Central Tendency, Variability, Skewness

Descriptive Statistics: Shape

• Central tendency and variability are useful to describe and summarise data.

• They cannot summarise asymmetry.

• Skewness is a measure of asymmetry (Calculating skewness will not be examined).

Skewness

• Symmetric distribution (skewness = 0): median = mean

• Right-skewed distribution (skewness > 0, positively skewed): median < mean

• Left-skewed distribution (skewness < 0, negatively skewed): median > mean

Summary for week 1

• Categorical data is summarised using tables and frequency counts, and visualised using histograms or pie charts.

• Distribution is the general shape that shows the probability that a random variable takes a certain value.

• Central tendency includes mean, mode (most commonly occurring value in an array of numbers) and median (the middle number if you sort the array)

  • Population mean: $μ = E(X) = \frac{x<em>1 + x</em>2 + \cdots + x<em>N}{N} = \frac{1}{N} \sum</em>{i=1}^{N} x_i$

  • Sample mean: $\bar{X} = \frac{x<em>1 + x</em>2 + \cdots + x<em>n}{n} = \frac{1}{n} \sum</em>{i=1}^{n} x_i$

• Variability includes variance, standard deviation and coefficient of variation

• Measure of shape: skewness

• Population Variance: $\sigma^2 = Var(X) = \frac{1}{N} \sum<em>{i=1}^{N} (x</em>i - \mu)^2$

• Sample Variance: $s^2 = \frac{1}{n-1} \sum<em>{i=1}^{n} (x</em>i - \bar{X})^2$

• Standard deviation: $\sigma = \sqrt{\frac{1}{N} \sum<em>{i=1}^{N} (x</em>i - \mu)^2}$ , $s = \sqrt{\frac{1}{n-1} \sum<em>{i=1}^{n} (x</em>i - \bar{X})^2}$

• Coefficient of variation: $\frac{\sigma}{\mu} \times 100\%$ , $\frac{s}{\bar{X}} \times 100\%$