Wednesday, October 9, 2024

Statistic

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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. 

Ans : 

10. 

Ans : 

11. Find the mean deviation about median for the following data :

Ans : 

12. Calculate the mean deviation about median age for the age distribution of 100 persons given below:

Ans : 

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

Ans :

4.

Ans : 

5.

Ans : 

6. Find the mean and standard deviation using short-cut method.

Ans : 

Find the mean and variance for the following frequency distributions in Exercises 7 and 8. 

7.

Ans : 

8. 

Ans : 

9. Find the mean, variance and standard deviation using short-cut method

Ans : 

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.] 

Ans : 

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