From de940be05e50206179bce346db85335b7a8cb0c4 Mon Sep 17 00:00:00 2001 From: eggy Date: Tue, 22 Sep 2020 22:19:46 -0400 Subject: [PATCH] math: add frequency data structures --- docs/mhf4u7.md | 59 ++++++++++++++++++++++++++++++++++++++++++++------ 1 file changed, 52 insertions(+), 7 deletions(-) diff --git a/docs/mhf4u7.md b/docs/mhf4u7.md index a5d129e..234a248 100644 --- a/docs/mhf4u7.md +++ b/docs/mhf4u7.md @@ -6,21 +6,21 @@ The course code for this page is **MHF4U7**. !!! note "Definition" - **Statistics:** The techniques and procedures to analyse, interpret, display, and make decisions based on data. - - **Descriptive statistics:** The use of methods to organise, display, and describe data by using various charts and summary methods to reduce data to a manageable size. + - **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.\ + - **Raw data:** Data collected prior to processing or ranking. ### Sampling A good sample: - represents the relevant features of the full population, - - is large enough so that it decently represents 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: @@ -47,13 +47,58 @@ The types of random sampling include: - **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 data set that lists ranges and the number of values in each range. It can be displayed using a frequency distribution table. +A **frequency distribution** is a table that lists categories/ranges and the number of values in each category/range. -!!! note "Definition" - +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. ## Resources