Grade 9 Statistics

Data: A collection of facts, such as numbers, words, measurements, observations, or just descriptions of things.
Example: The marks obtained by students in a test, the heights of students in a class.

Raw Data: Data collected in its original form before any organization or analysis.
Example: A list of test scores: 7, 9, 5, 8, 7, 10, 6, 9, 8, 7.

Types of Data (based on collection):

• **Primary Data:** Data collected directly by the investigator for a specific purpose (e.g., conducting a survey yourself).
• **Secondary Data:** Data collected from a source that already exists (e.g., using data from a government report or website).

Need for Organization: Raw data is often difficult to understand. Organizing data makes it easier to analyze and interpret.

Frequency: The number of times a particular observation occurs in a data set.

Frequency Distribution Table: A table that shows the frequency of each observation or class interval in a data set. It usually has columns for:

• Observation or Class Interval
• Tally Marks (for counting)
• Frequency (count)

Example: Raw data of marks: 7, 9, 5, 8, 7, 10, 6, 9, 8, 7.
Observations: 5, 6, 7, 8, 9, 10.

Step 1: List the distinct observations in ascending order.
5, 6, 7, 8, 9, 10.

Step 2: Go through the raw data and put a tally mark (|) next to the corresponding observation for each data point. Group tally marks in fives (|||| with a diagonal line across).
5: |, 6: |, 7: |||, 8: ||, 9: ||, 10: |

Step 3: Count the tally marks for each observation to get the frequency.
5: 1, 6: 1, 7: 3, 8: 2, 9: 2, 10: 1.

Step 4: Create the table with columns: Marks, Tally Marks, Frequency.
Marks | Tally Marks | Frequency
----- | ----------- | ---------
5 | | | 1
6 | | | 1
7 | ||| | 3
8 | || | 2
9 | || | 2
10 | | | 1
Total | | 10 (Sum of frequencies should equal total data points)

Definition: The mean (or average) of a data set is the sum of all observations divided by the total number of observations.

Formula: Mean = (Sum of all observations) / (Total number of observations)
In symbols, Mean = $\Sigma x / n$, where $\Sigma x$ is the sum of all observations and $n$ is the total number of observations.

Example Problem: Find the mean of the following data: 10, 20, 30, 40, 50.

Step 1: Sum all the observations.
Sum = 10 + 20 + 30 + 40 + 50 = 150.

Step 2: Count the total number of observations.
Number of observations (n) = 5.

Step 3: Use the formula: Mean = Sum / n.
Mean = 150 / 5

Step 4: Calculate the mean.
Mean = 30.

Result: The mean of the given data is 30.

Definition: The median is the middle observation in a data set that has been arranged in ascending or descending order. It divides the data into two equal halves.

Step 1: Arrange the data in ascending or descending order.
Example Data: 15, 10, 25, 20, 30.
Arranged: 10, 15, 20, 25, 30.

Step 2: Count the number of observations (n).
n = 5.

Step 3 (for odd n): If n is odd, the median is the ((n+1)/2)th observation.
For n=5, the median is the ((5+1)/2)th = 3rd observation.

Step 4 (for odd n): Find the value of the median observation in the arranged data.
Arranged data: 10, 15, **20**, 25, 30. The 3rd observation is 20.

Result (for odd n): The median is 20.

Step 3 (for even n): If n is even, the median is the average of the (n/2)th and the ((n/2)+1)th observations.
Example Data: 10, 15, 20, 25. n=4.
The median is the average of the (4/2)th = 2nd and the ((4/2)+1)th = 3rd observations.

Step 4 (for even n): Find the values of the two middle observations in the arranged data and calculate their average.
Arranged data: 10, **15**, **20**, 25. The 2nd observation is 15, the 3rd is 20.
Median = (15 + 20) / 2 = 35 / 2 = 17.5.

Result (for even n): The median is 17.5.

Definition: The mode is the observation that occurs most frequently in a data set. A data set can have one mode (unimodal), more than one mode (multimodal), or no mode.

Step 1: Examine the data set to find the frequency of each observation. A frequency distribution table (as in Example 2) is helpful for this.
Example Data: 7, 9, 5, 8, 7, 10, 6, 9, 8, 7.

Step 2: Identify the observation(s) with the highest frequency.
Frequencies: 5 (1), 6 (1), 7 (3), 8 (2), 9 (2), 10 (1).
The highest frequency is 3, which corresponds to the observation 7.

Result: The mode of the given data is 7.

Example 2 (Multimodal): Data: 1, 2, 2, 3, 3, 4.
Frequencies: 1 (1), 2 (2), 3 (2), 4 (1).
The highest frequency is 2, which corresponds to both 2 and 3.
Result: The modes are 2 and 3 (bimodal).

Example 3 (No Mode): Data: 1, 2, 3, 4, 5.
Frequencies: 1 (1), 2 (1), 3 (1), 4 (1), 5 (1).
All observations have the same frequency.
Result: There is no mode.