Concept Notes 1: Raw Data, Organization, Tally Marks & The Arithmetic Mean
Mathematics is not merely the study of static numbers; it is the science of discovering patterns and structure within information. When observations are collected for an investigation, they form data.
1. What is Data?
- Data: A collection of numerical facts, measurements, or observations gathered to draw conclusions or make decisions.
- Raw Data: Data collected in its original, unorganized form.
- Arrayed Data: Data arranged in ascending or descending numerical order.
2. Organizing Data: The Tally Frequency Table
When dealing with large numbers of observations, listing them as raw data leads to errors and confusion. We organize raw data into a Frequency Distribution Table:
- Observation / Variable ($x$): The distinct numerical value or category being measured.
- Tally Marks: Vertical tally strokes $|, ||, |||, ||||$ where every fifth occurrence is grouped by a diagonal slash $\cancel{||||}$ across the previous four.
- Frequency ($f$): The total number of times a particular observation occurs in the dataset.
Example 1: Organizing Test Scores
Suppose 20 students in Class 7 scored the following marks out of 10 in a mental math quiz:
| Marks ($x$) | Tally Marks | Frequency ($f$) |
|---|---|---|
| 5 | $||$ | 2 |
| 6 | $|||$ | 3 |
| 7 | $||||$ | 4 |
| 8 | $\cancel{||||} \ |$ | 6 |
| 9 | $|||$ | 3 |
| 10 | $||$ | 2 |
| Total | $N = 20$ |
At a glance, we can see that mark 8 was scored by the greatest number of students (6 students).
3. Representative Values (Central Tendency)
A single summary number that represents the central location or general characteristic of an entire dataset is called a Representative Value or Measure of Central Tendency. The three primary measures are:
- Arithmetic Mean
- Median
- Mode
4. The Arithmetic Mean
The Arithmetic Mean (often simply called the Mean or Average) is the most common representative value. It is defined as:
In mathematical notation:
Key Mathematical Properties of the Mean:
- Boundedness: The mean always lies strictly between the minimum observation and the maximum observation of the dataset:
The mean can never be smaller than the lowest value or larger than the highest value.
- Equilibrium (Zero Deviation): The sum of the deviations of each observation from the mean is identically zero:
- Change in Scale / Origin:
- If a constant $k$ is added to every observation, the new mean becomes $\bar{x} + k$.
- If every observation is multiplied by a constant $c$, the new mean becomes $c \cdot \bar{x}$.
5. Mean of Grouped / Frequency Data
When data is organized with frequencies ($f_1, f_2, \dots, f_k$), the sum of all values is $\sum (f_i \cdot x_i)$:
Using our test score table above:
The average score of the class is $7.55$ out of 10.