Statistics — mean, median, mode and standard deviation (KCSE Mathematics Form 3)

TL;DR

You'll learn how to calculate the mean, median, mode, and standard deviation, which are key ways to understand a set of numbers. These measures help you describe the 'center' and 'spread' of your data. Mastering them is crucial for interpreting data in your KCSE exams and beyond.

1. The Mental Model

Imagine you have a pile of different-sized stones. The mean is like finding the average weight if you distributed the total weight evenly. The median is the weight of the stone exactly in the middle if you lined them up from lightest to heaviest. The mode is the weight of the stone you have most often. Standard deviation tells you how spread out the weights are from the average.

2. The Core Material

Understanding Data: Raw vs. Grouped

Before we dive into calculations, it's important to distinguish between raw data and grouped data.
* Raw Data: This is data collected in its original form, like a list of individual scores: 12, 15, 10, 15, 18.
* Grouped Data: This is data organised into classes or intervals, often with frequencies, like a frequency distribution table. This makes large datasets easier to manage.

Mean (Average)

The mean is the sum of all values divided by the number of values.

For Raw Data

Formula: Mean ($\bar{x}$) = $\frac{\sum x}{n}$
Where:
* $\sum x$ is the sum of all data values.
* $n$ is the total number of data values.

Example: Find the mean of 5, 8, 12, 15, 5.
$\sum x = 5 + 8 + 12 + 15 + 5 = 45$
$n = 5$
Mean = $\frac{45}{5} = 9$

For Grouped Data (Using Assumed Mean Method)

When you have grouped data, especially with many classes, calculating the exact mean can be tedious. The assumed mean method simplifies this.

Formula: Mean ($\bar{x}$) = $A + \frac{\sum fd}{\sum f}$
Where:
* $A$ is the assumed mean (choose the midpoint of a class with a high frequency, usually near the middle).
* $f$ is the frequency of each class.
* $d$ is the deviation of the class midpoint ($x$) from the assumed mean ($d = x - A$).
* $\sum fd$ is the sum of (frequency $\times$ deviation) for all classes.
* $\sum f$ is the total frequency (total number of data points).

Steps for Assumed Mean:
1. Find the midpoint ($x$) for each class interval.
2. Choose an assumed mean ($A$) from one of the midpoints.
3. Calculate the deviation ($d = x - A$) for each midpoint.
4. Calculate $fd$ for each class.
5. Sum $fd$ and sum $f$.
6. Apply the formula.

Median (Middle Value)

The median is the middle value in an ordered dataset.

For Raw Data

1. Arrange the data in ascending order.
2. If $n$ is odd, the median is the value at the $\frac{n+1}{2}$ position.
3. If $n$ is even, the median is the average of the two middle values (at positions $\frac{n}{2}$ and $\frac{n}{2} + 1$).

Example (Odd $n$): Find the median of 5, 8, 12, 15, 5.
Ordered: 5, 5, 8, 12, 15
$n = 5$. Position = $\frac{5+1}{2} = 3^{rd}$. Median = 8.

Example (Even $n$): Find the median of 5, 8, 12, 15, 5, 10.
Ordered: 5, 5, 8, 10, 12, 15
$n = 6$. Positions = $\frac{6}{2} = 3^{rd}$ and $\frac{6}{2} + 1 = 4^{th}$.
Values are 8 and 10. Median = $\frac{8+10}{2} = 9$.

For Grouped Data

1. Calculate the cumulative frequency.
2. Find the median position: $\frac{\sum f}{2}$.
3. Identify the median class (the first class whose cumulative frequency is greater than or equal to the median position).
4. Apply the formula:
   Formula: Median = $L + \left(\frac{\frac{n}{2} - C_b}{f_m}\right) \times w$
   Where:
   • $L$ is the lower class boundary of the median class.
   • $n$ is the total frequency ($\sum f$).
   • $C_b$ is the cumulative frequency of the class before the median class.
   • $f_m$ is the frequency of the median class.
   • $w$ is the class width of the median class.

Mode (Most Frequent Value)

The mode is the value that appears most often in a dataset.

For Raw Data

Simply count the occurrences of each value.
Example: For 5, 8, 12, 15, 5, 10. The value 5 appears twice, others once. Mode = 5.
A dataset can have one mode (unimodal), multiple modes (multimodal), or no mode if all values appear with the same frequency.

For Grouped Data (Modal Class)

For grouped data, we usually identify the modal class, which is the class with the highest frequency.
Formula: Mode = $L + \left(\frac{f_m - f_1}{(f_m - f_1) + (f_m - f_2)}\right) \times w$
Where:
* $L$ is the lower class boundary of the modal class.
* $f_m$ is the frequency of the modal class.
* $f_1$ is the frequency of the class before the modal class.
* $f_2$ is the frequency of the class after the modal class.
* $w$ is the class width of the modal class.

Standard Deviation (Spread of Data)

Standard deviation measures the average amount of variability or dispersion around the mean. A low standard deviation means data points are generally close to the mean; a high standard deviation means data points are spread out over a wider range.

For Raw Data

Formula: Standard Deviation ($\sigma$) = $\sqrt{\frac{\sum (x - \bar{x})^2}{n}}$ or $\sqrt{\frac{\sum x^2}{n} - (\bar{x})^2}$
Where:
* $x$ is each individual data value.
* $\bar{x}$ is the mean.
* $n$ is the total number of data values.

Steps:
1. Calculate the mean ($\bar{x}$).
2. Subtract the mean from each data point ($x - \bar{x}$).
3. Square each result ($(x - \bar{x})^2$).
4. Sum these squared differences ($\sum (x - \bar{x})^2$).
5. Divide by $n$.
6. Take the square root.

For Grouped Data

Formula: Standard Deviation ($\sigma$) = $\sqrt{\frac{\sum fd^2}{\sum f} - \left(\frac{\sum fd}{\sum f}\right)^2}$
Where:
* $f$ is the frequency of each class.
* $d$ is the deviation of the class midpoint ($x$) from the assumed mean ($d = x - A$).
* $\sum fd^2$ is the sum of (frequency $\times$ deviation squared).
* $\sum fd$ is the sum of (frequency $\times$ deviation).
* $\sum f$ is the total frequency.

Steps:
1. Calculate midpoints ($x$) for each class.
2. Choose an assumed mean ($A$).
3. Calculate deviations ($d = x - A$).
4. Calculate $fd$.
5. Calculate $d^2$.
6. Calculate $fd^2$.
7. Sum $f$, $fd$, and $fd^2$.
8. Apply the formula.

graph TD
    A[Start] --> B{Data Type?};
    B -- Raw Data --> C[Calculate Mean: Sum values / Count];
    B -- Raw Data --> D[Calculate Median: Order data, find middle];
    B -- Raw Data --> E[Calculate Mode: Find most frequent value];
    B -- Raw Data --> F[Calculate Std Dev: Use raw data formula];
    B -- Grouped Data --> G[Calculate Midpoints (x)];
    G --> H[Choose Assumed Mean (A)];
    H --> I[Calculate Deviation (d = x - A)];
    I --> J[Calculate fd];
    J --> K[Calculate Mean: A + (Sum fd / Sum f)];
    J --> L[Calculate Cumulative Frequency];
    L --> M[Calculate Median: L + ((n/2 - Cb) / fm) * w];
    I --> N[Identify Modal Class (highest f)];
    N --> O[Calculate Mode: L + ((fm - f1) / ((fm - f1) + (fm - f2))) * w];
    J --> P[Calculate d^2];
    P --> Q[Calculate fd^2];
    Q --> R[Calculate Std Dev: Sqrt((Sum fd^2 / Sum f) - (Sum fd / Sum f)^2)];
    C & D & E & F & K & M & O & R --> Z[End];

3. Worked Example

Let's use the following grouped data representing the scores of 40 students in a math test:

1. Prepare the table with midpoints, deviations, etc.
Let's choose an assumed mean ($A$) of 34.5 (midpoint of the 30-39 class).
Class width ($w$) = 10 (e.g., 19.5 - 9.5 or 20 -