Class 10 Statistics Notes
Mean
The mean (average) of a set of numbers is calculated by summing all the numbers and dividing by the total count:
Mean = Sum of all numbers / Total count of numbers
Example:
Consider the following data set: 5, 8, 10, 12, 15
To find the mean:
Mean = (5 + 8 + 10 + 12 + 15) / 5
Mean = 50 / 5
Mean = 10
So, the mean of the given data set is 10.
Mode
The mode is the number that appears most frequently in a data set. It is possible to have one mode, more than one mode (bimodal, trimodal), or no mode (no repeated numbers).
Example:
Consider the following data set: 3, 5, 7, 5, 9, 5, 11
To find the mode:
Mode = 5
Here, the number 5 appears most frequently (three times) in the data set, making it the mode.
Median
The median is the middle value of a data set when it is arranged in ascending or descending order.
Example:
Consider the following data set: 4, 6, 9, 3, 2, 7
To find the median:
Step 1: Arrange the data in ascending order: 2, 3, 4, 6, 7, 9
Step 2: Since there are 6 numbers, the median will be the average of the two middle numbers:
Median = (4 + 6) / 2
Median = 10 / 2
Median = 5
So, the median of the given data set is 5.
Mean, Mode, and Median with Class Intervals
When dealing with class intervals, finding the mean, mode, and median requires additional calculations:
Let's consider the following data set with class intervals:
| Class Interval | Frequency |
|---|---|
| 10 - 20 | 4 |
| 20 - 30 | 6 |
| 30 - 40 | 10 |
| 40 - 50 | 8 |
| 50 - 60 | 5 |
Step 1: Calculate the midpoint of each class interval.
Step 2: Calculate the "Sum of (Midpoint × Frequency)" to find the mean.
Step 3: Identify the class interval with the highest frequency to find the mode.
Step 4: Calculate the cumulative frequency and find the median.
Let's calculate each of these measures:
Step 1: Calculate the midpoint of each class interval.
| Class Interval | Midpoint |
|---|---|
| 10 - 20 | 15 |
| 20 - 30 | 25 |
| 30 - 40 | 35 |
| 40 - 50 | 45 |
| 50 - 60 | 55 |
Step 2: Calculate the "Sum of (Midpoint × Frequency)" to find the mean.
Mean (𝑋̄) = (Σ(𝑋 × 𝑓)) / Σ𝑓
Where Σ(𝑋 × 𝑓) is the sum of (Midpoint × Frequency) and Σ𝑓 is the total frequency.
Mean (𝑋̄) = ((15 × 4) + (25 × 6) + (35 × 10) + (45 × 8) + (55 × 5)) / (4 + 6 + 10 + 8 + 5)
Mean (𝑋̄) = (60 + 150 + 350 + 360 + 275) / 33
Mean (𝑋̄) = 1195 / 33 ≈ 36.21 (rounded to two decimal places)
So, the mean of the given data set is approximately 36.21.
Step 3: Identify the class interval with the highest frequency to find the mode.
The class interval with the highest frequency is "30 - 40" with a frequency of 10. So, the mode is 35 (midpoint of the mode class interval).
Step 4: Calculate the cumulative frequency and find the median.
The cumulative frequency is the running total of frequencies as we go through the classes. It helps us find the median.
| Class Interval | Frequency | Cumulative Frequency |
|---|---|---|
| 10 - 20 | 4 | 4 |
| 20 - 30 | 6 | 10 |
| 30 - 40 | 10 | 20 |
| 40 - 50 | 8 | 28 |
| 50 - 60 | 5 | 33 |
The median is the value that falls in the middle when the data is arranged in ascending order. Since we have a total of 33 data points (odd), the median will be the value corresponding to the 17th data point.
The 17th data point falls in the class interval "30 - 40". To find the exact median, we can use the following formula:
Median = L + ((n/2 - F) / f) × w
Where:
L = Lower boundary of the median class interval (30)
n = Total number of data points (33)
F = Cumulative frequency of the class interval before the median class interval (10)
f = Frequency of the median class interval (10)
w = Width of the class interval (40 - 30 = 10)
Median = 30 + ((33/2 - 10) / 10) × 10
Median = 30 + (16.5 - 10) ≈ 36.5
So, the median of the given data set is approximately 36.5.
Class Interval Mode Example
Example: Class Interval Data Set
Consider the following data set with class intervals and their respective frequencies:
| Class Interval | Frequency |
|---|---|
| 10 - 20 | 4 |
| 20 - 30 | 6 |
| 30 - 40 | 10 |
| 40 - 50 | 8 |
| 50 - 60 | 5 |
Finding the Mode
To find the mode, we use the formula:
Mode = L + ((f0 - f1) / (2 * f0 - f1 - f2)) * h
where:
L = Lower boundary of the modal class interval
f0 = Frequency of the modal class interval (30 - 40)
f1 = Frequency of the class interval before the modal class interval (20 - 30)
f2 = Frequency of the class interval after the modal class interval (40 - 50)
h = Width of the class interval (40 - 30 = 10)
Using the provided data:
L = 30
f0 = 10
f1 = 6
f2 = 8
h = 10
Mode = 30 + ((10 - 6) / (2 * 10 - 6 - 8)) * 10
Mode = 30 + (4 / (20 - 6 - 8)) * 10
Mode = 30 + (4 / 6) * 10
Mode = 30 + (0.67) * 10
Mode ≈ 30 + 6.7
Mode ≈ 36.7
Therefore, the mode of the given data set is approximately 36.7.
Class Interval Mean Example
Example: Class Interval Data Set
Consider the following data set with class intervals and their respective frequencies:
| Class Interval | Frequency |
|---|---|
| 10 - 20 | 4 |
| 20 - 30 | 6 |
| 30 - 40 | 10 |
| 40 - 50 | 8 |
| 50 - 60 | 5 |
Finding the Mean
To find the mean, we use the formula:
Mean = A + ((Σ(fᵢ * uᵢ)) / Σfᵢ) * h
where:
A = Assumed mean (we will take it as the midpoint of the data)
h = Width of the class interval (40 - 30 = 10)
Step 1: Calculate the assumed mean (A) by taking the midpoint of the data:
A = (10 + 20) / 2 = 15
Step 2: Calculate the deviation (uᵢ) of the midpoint of each class interval from the assumed mean (A):
| Class Interval | Midpoint (mᵢ) | Deviation (uᵢ) | Frequency (fᵢ) | fᵢ * uᵢ |
|---|---|---|---|---|
| 10 - 20 | 15 | 0 | 4 | 0 |
| 20 - 30 | 25 | 10 | 6 | 60 |
| 30 - 40 | 35 | 20 | 10 | 200 |
| 40 - 50 | 45 | 30 | 8 | 240 |
| 50 - 60 | 55 | 40 | 5 | 200 |
Step 3: Calculate the sum of (fᵢ * uᵢ):
Σ(fᵢ * uᵢ) = 0 + 60 + 200 + 240 + 200 = 700
Step 4: Calculate the sum of frequencies (Σfᵢ):
Σfᵢ = 4 + 6 + 10 + 8 + 5 = 33
Step 5: Calculate the mean:
Mean = A + ((Σ(fᵢ * uᵢ)) / Σfᵢ) * h
Mean = 15 + (700 / 33) * 10
Mean ≈ 15 + 212.12
Mean ≈ 227.12 (rounded to two decimal places)
Therefore, the mean of the given data set is approximately 227.12.