Chapter 13: Statistics
Introduction:
- It helps us make informed decisions and understand patterns in data.
Measures of Central Tendency:
- Mean: The average value of a dataset.
- Median.
- Mode.
Measures of Dispersion:
- Range
- Variance.
- Standard Deviation: The square root of the variance.
Probability:
- Probability.
- Probability of an Event.
- Probability of Complementary Events: P(A’) = 1 – P(A).
- Conditional Probability: The probability of event B occurring given that event A has already occurred.
Binomial Distribution:
- Binomial Experiment: An experiment with a fixed number of trials, each trial having only two possible outcomes (success or failure), and the trials are independent.
- Binomial Probability: The probability of getting exactly k successes in n trials.
Other Topics:
- Frequency Distribution: A table or graph showing the frequency of each value in a dataset.
- Cumulative Frequency Distribution: A table or graph showing the cumulative frequency of each value in a dataset.
- Histogram: A bar graph where the bars are touching.
- Ogive: A graph of the cumulative frequency distribution.
- Correlation and Regression: The relationship between two variables.
Key Concepts:
- Central tendency and dispersion
- Probability
- Binomial distribution
- Frequency distributions
- Correlation and regression
Exercise 13.1
Find the mean deviation about the mean for the data in Exercises 1 and 2.
1. 4, 7, 8, 9, 10, 12, 13, 17
2. 38, 70, 48, 40, 42, 55, 63, 46, 54, 44
Ans :
1: 4, 7, 8, 9, 10, 12, 13, 17
Step 1: Calculate the mean:
Mean (x̄) = (4 + 7 + 8 + 9 + 10 + 12 + 13 + 17) / 8 = 96 / 8 = 12
Step 2: Calculate the absolute deviations:
| xi – x̄ |
| 4 – 12 | = 8
| 7 – 12 | = 5
| 8 – 12 | = 4
| 9 – 12 | = 3
| 10 – 12 | = 2
| 12 – 12 | = 0
| 13 – 12 | = 1
| 17 – 12 | = 5
Step 3: Calculate the mean deviation:
Mean Deviation = (8 + 5 + 4 + 3 + 2 + 0 + 1 + 5) / 8 = 28 / 8 = 3.5
2.
Step 1: Calculate the mean:
Mean (x̄) = (38 + 70 + 48 + 40 + 42 + 55 + 63 + 46 + 54 + 44) / 10
= 500 / 10
= 50
Step 2: Calculate the absolute deviations from the mean:
| xi – x̄ |
| 38 – 50 | = 12
| 70 – 50 | = 20
| 48 – 50 | = 2
| 40 – 50 | = 10
| 42 – 50 | = 8
| 55 – 50 | = 5
| 63 – 50 | = 13
| 46 – 50 | = 4
| 54 – 50 | = 4
| 44 – 50 | = 6
Step 3: Calculate the mean deviation:
Mean Deviation = (Σ|xi – x̄|) / n
= (12 + 20 + 2 + 10 + 8 + 5 + 13 + 4 + 4 + 6) / 10
= 84 / 10
= 8.4
Find the mean deviation about the median for the data in Exercises 3 and 4.
3. 13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17
4. 36, 72, 46, 42, 60, 45, 53, 46, 51, 49
Ans :
3.
Step 1: Arrange the data in ascending order:
10, 11, 11, 12, 13, 13, 14, 16, 16, 17, 17, 18
Step 2: Calculate the median:
Since there are 12 data points, the median is the average of the 6th and 7th terms.
Median = (13 + 13) / 2 = 13
Step 3: Calculate the absolute deviations from the median:
| xi – Median |
| 10 – 13 | = 3 | 11 – 13 | = 2 | 11 – 13 | = 2 | 12 – 13 | = 1 | 13 – 13 | = 0 | 13 – 13 | = 0 | 14 – 13 | = 1 | 16 – 13 | = 3 | 16 – 13 | = 3 | 17 – 13 | = 4 | 17 – 13 | = 4 | 18 – 13 | = 5
Step 4: Calculate the mean deviation about the median:
Mean Deviation = (Σ|xi – Median|) / n
= (3 + 2 + 2 + 1 + 0 + 0 + 1 + 3 + 3 + 4 + 4 + 5) / 12
= 28 / 12
= 7/3
4.
Step 1: Arrange the data in ascending order:
36, 42, 45, 46, 46, 49, 51, 53, 60, 72
Step 2: Calculate the median:
Since there are 10 data points, the median is the average of the 5th and 6th terms.
Median = (46 + 49) / 2 = 47.5
Step 3: Calculate the absolute deviations from the median:
| xi – Median |
| 36 – 47.5 | = 11.5 | 42 – 47.5 | = 5.5 | 45 – 47.5 | = 2.5 | 46 – 47.5 | = 1.5 | 46 – 47.5 | = 1.5 | 49 – 47.5 | = 1.5 | 51 – 47.5 | = 3.5 | 53 – 47.5 | = 5.5 | 60 – 47.5 | = 12.5 | 72 – 47.5 | = 24.5
Step 4: Calculate the mean deviation about the median:
Mean Deviation = (Σ|xi – Median|) / n
= (11.5 + 5.5 + 2.5 + 1.5 + 1.5 + 1.5 + 3.5 + 5.5 + 12.5 + 24.5) / 10
= 70 / 10
= 7
Find the mean deviation about the mean for the data in Exercises 5 and 6. 5.
5. Xi 5 10 15 20 25
f i 7 4 6 3 5
6. Xi 10 30 50 70 90
f i 4 24 28 16 8
Ans :
5.
6.
Find the mean deviation about the median for the data in Exercises 7 and 8. 7.
7. Xi 5 7 9 10 12 15
f i 8 6 2 2 2 6
8. xi 15 21 27 30 35
f i 3 5 6 7 8
Ans :
7.
8.
9. Find the mean deviation about the mean for the data in Exercises 9 and 10.
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10.
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11. Find the mean deviation about median for the following data :
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12. Calculate the mean deviation about median age for the age distribution of 100 persons given below:
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Exercise 13.2
Find the mean and variance for each of the data in Exercies 1 to 5.
1. 6, 7, 10, 12, 13, 4, 8, 12
Ans :
Mean = (Σxi) / n
Variance = (Σ(xi – x̄)^2) / n
where:
- Σ is the summation symbol
- xi is the ith data point
- x̄ is the mean of the data
Mean = (6 + 7 + 10 + 12 + 13 + 4 + 8 + 12) / 8 = 72 / 8 = 9
Variance = ((6-9)^2 + (7-9)^2 + (10-9)^2 + (12-9)^2 + (13-9)^2 + (4-9)^2 +(8-9)^2 + (12-9)^2) / 8 = 74 / 8
= 9.25
2. First n natural numbers
Ans :
Mean:
Mean = (n + 1) / 2
Variance:
Variance = (n^2 – 1) / 12
These formulas are derived from the properties of arithmetic series.
Therefore, the mean of the first n natural numbers is (n + 1) / 2, and the variance is (n^2 – 1) / 12.
3. First 10 multiples of 3
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4.
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5.
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6. Find the mean and standard deviation using short-cut method.
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Find the mean and variance for the following frequency distributions in Exercises 7 and 8.
7.
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8.
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9. Find the mean, variance and standard deviation using short-cut method
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10. The diameters of circles (in mm) drawn in a design are given below:
Calculate the standard deviation and mean diameter of the circles. [ Hint First make the data continuous by making the classes as 32.5-36.5, 36.5-40.5, 40.5-44.5, 44.5 – 48.5, 48.5 – 52.5 and then proceed.]
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