math: add frequency data structures

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. 
+
+<img src="/resources/images/frequency-discrete.png" width=700>(Source: Kognity)</img>
+
+!!! 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