5.4 KiB
SL Math - Analysis and Approaches - A
The course code for this page is MHF4U7.
4 - Statistics and probability
!!! note “Definition” - Statistics: The techniques and procedures to analyse, interpret, display, and make decisions based on data. - Descriptive statistics: The use of methods to work with and describe the entire data set. - Inferential statistics: The use of samples to make judgements about a population. - Data set: A collection of data with elements and observations, typically in the form of a table. It is similar to a map or dictionary in programming. - Element: The name of an observation(s), similar to a key to a map/dictionary in programming. - Observation: The collected data linked to an element, similar to a value to a map/dictionary in programming. - Population: A collection of all elements of interest within a data set. - Sample: The selection of a few elements within a population to represent that population. - Raw data: Data collected prior to processing or ranking.
Sampling
A good sample:
- represents the relevant features of the full population,
- is as large as reasonably possible so that it decently represents the full population,
- and is random.
The types of random sampling include:
- Simple: Choosing a sample completely randomly.
- Convenience: Choosing a sample based on ease of access to the data.
- Systematic: Choosing a random starting point, then choosing the rest of the sample at a consistent interval in a list.
- Quota: Choosing a sample whose members have specific characteristics.
- Stratified: Choosing a sample so that the proportion of specific characteristics matches that of the population.
??? example - Simple: Using a random number generator to pick items from a list. - Convenience: Asking the first 20 people met to answer a survey, - Systematic: Rolling a die and getting a 6, so choosing the 6th element and every 10th element after that. - Quota: Ensuring that all members of the sample all wear red jackets. - Stratified: The population is 45% male and 55% female, so the proportion of the sample is also 45% male and 55% female.
Types of data
!!! note “Definition” - Quantitative variable: A variable that is numerical and can be sorted. - Discrete variable: A quantitative variable that is countable. - Continuous variable: A quantitative variable that can contain an infinite number of values between any two values. - Qualitative variable: A variable that is not numerical and cannot be sorted. - Bias: An unfair influence in data during the collection process, causing the data to be not truly representative of the population.
Frequency distribution
A frequency distribution is a table that lists categories/ranges and the number of values in each category/range.
A frequency distribution table includes:
- A number of classes, all of the same width.
- This number is arbitrarily chosen, but a commonly used formula is \(\lceil\sqrt{\text{# of elements}}\rceil\).
- The width (size) of each class is \(\lceil\frac{\text{max value} - \text{min value}}{\text{number of classes}}\rceil\).
- Each class includes its lower bound and excludes its upper bound (\(\text{lower} ≤ x < \text{upper}\))
- The relative frequency of a data set is the percentage of the whole data set present in that class in decimal form.
- The number of values that fall under each class.
- The largest value can either be included in the final class (changing its range to \(\text{lower} ≤ x ≤ \text{highest}\)), or put in a completely new class above the largest class.
??? example | Height \(x\) (cm) | Frequency | | — | — | | \(1≤x<5\) | 2 | | \(5≤x<9\) | 3 | | \(9≤x≤14\) | 1 |
For a given class \(i\), the midpoint of that class is as follows: \[x_{i} = \frac{\text{lower bound} + \text{upper bound}}{2}\]
Representing frequency
A stem and leaf plot can list out all the data points while grouping them simultaneously.
A frequency histogram can be used to represent frequency distribution, with the x-axis containing class boundaries, and the y-axis representing frequency.
(Source:
Kognity)
!!! note If data is discrete, a gap must be left between the bars. If data is continuous, there must not be a gap between the bars.
A cumulative frequency table can be used to find the number of data values below a certain class boundary. It involves the addition of a cumulative frequency column which represents the sum of the frequency of the current class as well as every class before it. It is similar to a prefix sum array in computer science.
??? example | Height \(h\) (cm) | Frequency | Cumulative frequency | | — | — | — | | \(1≤h<10\) | 2 | 2 | | \(10≤h<19\) | 5 | 7 |
Outliers
Outliers are data values that significantly differs from the rest of the data set. They may be because of:
- a random natural occurrence, or
- abnormal circumstances
Outliers can be ignored once identified.