Statistics Basics Quiz

Mathematics of Data Management

• Part 1 covers Descriptive Statistics, making up 30% of the course.
• Course notes are from Lower Canada College.

Unit 1 - Introduction to Statistics

• A. Basics
  • A survey is a process of gathering information for informed decisions.
  • Data are observations (e.g., eye color, salary, height).
  • A population is the complete group.
  • A sample is a subset of the population.
  • Sampling Techniques
    • Voluntary Response Sample: Participants decide whether to participate. This method has limitations, as the sample may not represent the population
    • Simple Random Sample: Participants are selected randomly. This approach can ensure the sample represents the population fairly. An example is selecting 10 students from each grade randomly.
  • Sources of Bias
    • Sampling Bias: When the sample doesn't reflect the characteristics of the whole population. An example is only surveying Montreal Canadiens fans to determine favourite NHL team.
    • Non-Response Bias: When specific groups aren't represented in a survey because they opted out of participating. This often arises when surveys are optional. An example is when only 50% of students respond to a survey on athletics.

Generating A Simple Random Sample

• Steps to generate a simple random sample using a calculator are described, including seeding the random number generator.

Types of Data

• Qualitative Data: Characterized by names or labels. Examples are eye color, political party affiliations.
• Quantitative Data: Characterized by numerical measurements.
  • Discrete Data: Finite or countable values. Examples include the number of eggs laid in a week, rolls of a die.
  • Continuous Data: Infinite possible values. An example is the amount of milk a cow yields in a year (any value between 0 to 7000 liters).

Levels of Measurement

• Nominal: Categories, no order (e.g., favorite food).
• Ordinal: Categories with an order (e.g., letter grade).
• Interval: Has meaningful differences between values but no true zero (e.g., temperature in Celsius).
• Ratio: Has meaningful differences and a true zero (e.g., salary, age).

C. Collecting Data

• Sampling Techniques
  • Systematic Sampling: Select participants at regular intervals. Good for large populations, but potentially prone to bias if the intervals have a hidden pattern
  • Stratified Sampling: Divides the population into strata (groups with shared characteristics). Random sampling within each strata. Can improve representation, but also requires significant effort
  • Cluster Sampling: Divide the population into clusters, randomly select clusters. Can be more efficient with large populations, but may reduce diversity.
  • Convenience Sampling: Selecting accessible participants. This method usually isn't reliable, as the sample is unlikely to represent the population truly.

Unit 2 - Graphing and Summarizing Data

• A. Graphing data
  • Data visualisation facilitates easier understanding and makes predictions.
  • Creating bar charts, pie charts, and Pareto charts is discussed.
  • Graphs organise and summarise data, allowing quicker analysis.
• Frequency Distribution Table
  • Tabulates data with frequencies, relative frequencies and cumulative frequencies
• Histograms
  • Bars are side by side, similar to bar graphs, where each bar shows frequency of data within a class interval
• Frequency Polygon
  • Connect the midpoints of adjacent bars
• Ogive (Cumulative Frequency Polygon) -Shows cumulative frequencies
• Stem and Leaf Diagrams -Visualises data by separating each data value into a stem and leaf

Measures of Central Tendency

• Mean: Average of the data values
• Median: Middle value when data is ordered
• Mode: Most frequent value

Measures of Dispersion (Spread)

• Range: Difference between highest and lowest data values.
• Variance: Average of the squared differences from the mean.
• Standard Deviation: Square root of the variance.

Finding the mean, variance and standard deviation

• Steps for calculating mean, variance and standard deviation are illustrated with an example of dog heights.
  • Calculate the mean (average) of the data values
  • Calculate the difference between each value and the mean, and square these differences
  • Calculate the average (mean) of the squared differences
  • Calculate the square root of the variance to get the standard deviation

Unit 3 - Probability

• A. Basics of Probability
  • Probability is about the likelihood of an event occurring. An example of an event is getting a boy or a girl.
  • An event is a group of outcomes from a particular procedure.
  • A simple event has no subsets. The sample space is the whole group of all the possible simple events.
  • Probability of an event is between zero and one.
• Types of Events
• Complementary Events: The events that do not occur. The complement of event A occurs if A doesn't occur
• Compound Events A compound event occurs if two or more simple events occur. An example is if both A and B occur
• Independent/Dependent Events: If the occurence of event one does not affect the probability of the other (indepedent), or if the occurrence of one event does affect the probability of the other (dependent).
• Conditional probability The probability of an event given some additional information, that some other event has already occurred.

Counting

• Permutations: Order matters.
• Combinations: Order does not matter.
• Rules for calculating them are given, for both when all items are different, or some items are the same.
  • Factorial Rule: Calculating permutations when there are the same number of items as options
  • Fundamental Counting Rule: Calculating possible outcomes of multiple events
  • Permutation Rule: Calculating permutations when there are multiple items that are identical

Unit 4 - The Normal Distribution

• A. Normal Distributions and standard deviations:

• Normal curves are symmetrical and bell shaped

• mean, median, mode are equal and centered in the distribution

• 68% of data values are within one standard deviation of the mean

• 95% are within two standard deviations of the mean

• 99.7% are within three standard deviations of the mean. -z-scores convert any normal distribution to a standard normal distribution. Standardized z-scores allow for comparisons between different distributions.

• Range Rule of Thumb: The vast majority of values live within 2 standard deviations of the mean

• B. Skewness:

  • Visual representation of how the distribution is lopsided or not
  • Measures of skewness quantify how symmetrical the distribution is

• C. Standard Normal distributions and z-scores:

  • Allows for comparison between different distributions

• E. Percentages and values (Normal Distribution):

  • Identifying specific values (e.g., heights) that fall within given percentiles of a normal distribution.

• F. Proving Normalcy:

  • Identify if distribution satisfies characteristics of a normal distribution

Test your understanding of fundamental statistical concepts including standard deviation, central tendency, and data summarization methods. This quiz covers essential aspects of statistics, making it perfect for beginners and intermediate learners alike.