Mathematics Statistics and Probability Syllabus

Statistics and Probability syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Statistics and Probability in Mathematics, not a summary of it.

1. 1

   Descriptive Statistics

   overview

   Examined as a single unit within Statistics and Probability — no further topic split in the official outline.

2. 2

   Probability Theory

   overview

   Examined as a single unit within Statistics and Probability — no further topic split in the official outline.

3. 3

   Random Variables

   overview

   Examined as a single unit within Statistics and Probability — no further topic split in the official outline.

4. 4

   Probability Distributions

   overview

   Examined as a single unit within Statistics and Probability — no further topic split in the official outline.

5. 5

   Sampling Distributions

   overview

   Examined as a single unit within Statistics and Probability — no further topic split in the official outline.

6. 6

   Inferential Statistics

   overview

   Examined as a single unit within Statistics and Probability — no further topic split in the official outline.

7. 7

   Hypothesis Testing

   overview

   Examined as a single unit within Statistics and Probability — no further topic split in the official outline.

8. 8

   Regression Analysis

   overview

   Examined as a single unit within Statistics and Probability — no further topic split in the official outline.

9. 9

   ANOVA

   overview

   Examined as a single unit within Statistics and Probability — no further topic split in the official outline.

Statistics and Probability flashcards for Mathematics

18 of 50 cards from the Statistics and Probability deck — real questions with worked answers.

1. What is the difference between a population and a sample in statistics?

   A population is the entire set of individuals or items of interest. A sample is a subset of the population actually selected and measured to make inferences about the population.

2. Distinguish between a parameter and a statistic.

   A parameter is a numerical summary of a population (e.g. population mean $\mu$). A statistic is a numerical summary of a sample (e.g. sample mean $\bar{x}$) used to estimate a parameter.

3. How is the mean (arithmetic average) of a data set computed?

   $$\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_{i}$$ the sum of all values divided by the number of values $n$.

4. How do you find the median of a data set?

   Order the data. If $n$ is odd, the median is the middle value; if $n$ is even, it is the average of the two middle values.

5. What is the mode of a data set?

   The value (or values) that occur most frequently. A data set may be unimodal, bimodal, multimodal, or have no mode.

6. Define the range of a data set.

   The difference between the maximum and minimum values: $\text{Range} = x_{\max} - x_{\min}$.

7. What is the formula for the population variance $\sigma^{2}$?

   $$\sigma^{2} = \frac{1}{N}\sum_{i=1}^{N}(x_{i}-\mu)^{2}$$ the mean of the squared deviations from the population mean $\mu$.

8. What is the formula for the sample variance $s^{2}$, and why is it divided by $n-1$?

   $$s^{2} = \frac{1}{n-1}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}$$ Dividing by $n-1$ (Bessel's correction) makes $s^{2}$ an unbiased estimator of $\sigma^{2}$.

9. How is the standard deviation related to the variance?

   The standard deviation is the square root of the variance: $\sigma = \sqrt{\sigma^{2}}$ (population) or $s = \sqrt{s^{2}}$ (sample). It has the same units as the data.

10. Define the interquartile range (IQR).

    $\text{IQR} = Q_{3} - Q_{1}$, the difference between the third (75th percentile) and first (25th percentile) quartiles. It measures the spread of the middle 50% of data.

11. What rule using the IQR identifies outliers?

    A value is an outlier if it lies below $Q_{1} - 1.5\,\text{IQR}$ or above $Q_{3} + 1.5\,\text{IQR}$.

12. What is a z-score and how is it calculated?

    A z-score gives the number of standard deviations a value lies from the mean: $$z = \frac{x - \mu}{\sigma}$$

13. State the empirical rule (68-95-99.7 rule) for a normal distribution.

    Approximately $68\%$ of data lies within $1\sigma$ of the mean, $95\%$ within $2\sigma$, and $99.7\%$ within $3\sigma$.

14. Compare the mean and median as measures of center with respect to skew.

    The mean is sensitive to outliers and skew; the median is resistant. In a right-skewed distribution mean > median; in a left-skewed distribution mean < median.

15. What is the difference between categorical (qualitative) and numerical (quantitative) data?

    Categorical data place items into groups/labels (e.g. color, gender). Numerical data are measured counts or quantities and can be discrete or continuous.

16. Distinguish discrete from continuous quantitative variables.

    Discrete variables take countable, separate values (e.g. number of children). Continuous variables can take any value in an interval (e.g. height, time).

17. Name the four levels of measurement.

    Nominal (labels), Ordinal (ordered categories), Interval (ordered, equal spacing, no true zero), and Ratio (interval with a true zero).

18. What is the classical (theoretical) definition of the probability of an event?

    For equally likely outcomes, $$P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$$

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