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Statistics · Class 12 Commerce

Ch 7Normal Distribution — Class 12 Statistics, concept-first.

Up to this point in the Gujarat Std-12 Statistics syllabus, the Binomial and Poisson distributions dealt with a discrete random variable — one that takes only specific, countable values (0, 1, 2, 3, …).

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Concepts

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Key concepts

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Properties of the Normal Distribution

The normal curve is bell-shaped and perfectly symmetric about its mean, so mean = median = mode. It is unimodal, asymptotic to the x-axis on both tails, has points of inflection at , and the total area under it equals 1…

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

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Continuous Probability Distributions — From Discrete to Continuous

Up to this point in the Gujarat Std-12 Statistics syllabus, the Binomial and Poisson distributions dealt with a discrete random variable — one that takes only specific, countable values (0, 1, 2, 3, ……

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The Normal Curve and Its Properties

A random variable is said to follow a normal distribution with mean and variance — written — if its probability density function is A Std-12 Commerce student is never asked to use this formula directl…

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Standardization: The Standard Normal Variate (Z)

In practice, business and economic data can have any mean and any standard deviation — wages might average ₹8,000 with an SD of ₹1,200, while exam marks average 55 with an SD of 10.

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Area Under the Normal Curve — Using the Standard Normal Table

Once a raw value has been standardized to , the Standard Normal Table gives — the cumulative area to the LEFT of under the standard normal curve, which is exactly .

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Applications of the Normal Distribution in Business and Statistics

The normal distribution is one of the most widely applied tools in business statistics precisely because so many real variables approximate it once observations are numerous.

Exercises

+Show 8 questions8 questions
  1. Q6State whether the following statements about the normal distribution are True or False, giving a brief reason for each: (i) The normal curve…Free
  2. Q7If $X \sim N(500, 64)$, find $P(X > 510)$.Free
  3. Q8The monthly wages of workers in a factory are normally distributed with a mean of ₹8,000 and a standard deviation of ₹1,200. Out of 500 work…Free
  4. Q9A normal distribution has mean 100 and standard deviation 15. Find the two values of X between which the middle 95% of the distribution lies…Preview
  5. Q10Explain briefly, using the Empirical Rule, what percentage of observations in a normal distribution lie within two standard deviations of th…Preview
  6. Q11Marks of 1,000 students in a Statistics examination are normally distributed with a mean of 55 and a standard deviation of 10. How many stud…Preview
  7. Q12In a standard normal distribution, what are the values of the mean and the variance? Choose the correct option: (a) mean = 1, variance = 0 (…Preview
  8. Q13Explain briefly why the probability of a continuous random variable (such as a normally distributed variable) is represented as the AREA und…Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 33 questions33 questions
  1. Q1In normal distribution usually which limits include 99% of the observations? (a) $\mu \pm 1.96\sigma$ (b) $\mu \pm 2\sigma$ (c) $\mu \pm 3\s…Preview
  2. Q2Mean and the first quartile for a normal distribution are $11$ and $3$ respectively. Which of the following is the value of the third quarti…Preview
  3. Q3For which value of standard normal variable, the standard normal curve is symmetric on both the sides?Preview
  4. Q4Define probability density function for normal variable.Preview
  5. Q5For a normal variable, mean deviation is $48$ and its third quartile is $120$. Estimate its first quartile.Preview
  6. Q6The weight of randomly selected 500 adult persons from a region of a city follows normal distribution. If the average weight of these person…Preview
  7. Q7Which of the following are mean and variance of standard normal variable? (a) Mean = 0, variance = 1 (b) Mean = 1, variance = 0 (c) Mean = 0…Preview
  8. Q8What is the area under the normal curve to the left hand side of perpendicular line $\mu$? (a) 0 (b) 0.5 (c) 1 (d) $-0.5$Preview
  9. Q9If the distribution of normal variable is shown as $N(20, 4)$ then which of the following intervals includes 99.73% of observations? (a) $(1…Preview
  10. Q10Give the values of the constants used in probability density function of normal variable.Preview
  11. Q11What percentage of area is covered under the normal curve within the range $\mu - 2\sigma$ to $\mu + 2\sigma$?Preview
  12. Q12(A) For a normal distribution, the first quartile and the mean deviation are 20 and 24 respectively. Obtain an estimate of the value of mode…Preview
  13. Q13The weight of randomly selected 500 adult persons from a region of a city follows normal distribution. The average weight of these persons i…Preview
  14. Q14What is the area under the normal curve to the left hand side of perpendicular line at $X = \mu$? (a) (A) $0$ (b) (B) $-0.5$ (c) (C) $1$ (d)…Preview
  15. Q15Which of the following is the approximate value of mean deviation for normal variable? (a) (A) $\frac{4}{5}\sigma$ (b) (B) $\frac{4}{5}\mu$…Preview
  16. Q16What is the shape of normal curve?Preview
  17. Q17Mean and the first quartile for a normal distribution are 11 and 3 respectively. Find the value of the third quartile.Preview
  18. Q18The probability density function of a normal variable $X$ is defined as under $f(x) = \text{constant} \cdot e^{-\frac{1}{2}\left(\frac{x - 2…Preview
  19. Q19The weight of randomly selected 500 adult persons from a region of a city follows normal distribution. The average weight of these persons i…Preview
  20. Q20Which of the following are mean and variance of standard normal variable? (a) Mean $= 0$, Variance $= 1$ (b) Mean $= 1$, Variance $= 0$ (c)…Preview
  21. Q21What is the area under the normal curve to the right hand side of perpendicular line at $X = \mu$ ? (a) $0$ (b) $0.5$ (c) $1$ (d) $-0.5$Preview
  22. Q22Mean and the first quartile for a normal distribution are 15 and 11 respectively. Which of the following is the value of the third quartile?…Preview
  23. Q23"Standard score is independent of unit of measurement". Is this statement true or false?Preview
  24. Q24The approximate value of mean deviation for a normal distribution is 12. Find its standard deviation.Preview
  25. Q25The average monthly expense of students residing in university hostel is ₹2,000 and its standard deviation is ₹500. If the monthly expense o…Preview
  26. Q26A normal distribution has mean 52 and variance 64. Obtain estimated limits which include exactly middle 60% of the observations. [Note : Bli…Preview
  27. Q27For a normal variable $X$ with mean $\mu$ and standard deviation $\sigma$, which of the following is standard normal variable $Z$ for it? (a…Preview
  28. Q28In normal distribution, usually which limits include 99% of the observations? (a) $\mu \pm 1.96\sigma$ (b) $\mu \pm 2\sigma$ (c) $\mu \pm 3\…Preview
  29. Q29For a normal distribution, approximate value of mean deviation is 20. Which of the following is the value of quartile deviation? (a) $\dfrac…Preview
  30. Q30What is the shape of normal curve?Preview
  31. Q31For a probability distribution of standard normal variable, state the estimated limits for the middle 50% observations.Preview
  32. Q32(A) This question is only for normal students. The weight of randomly selected 500 adult persons from a region of a city follows normal dist…Preview
  33. Q33(A) This question is only for normal students. The profit in daily business of a businessman having grocery shop follows normal distribution…Preview

More questions

+Show 5 questions5 questions
  1. Example 1If $X \sim N(50, 100)$ — i.e. a normal distribution with mean 50 and variance 100 — find the standard normal variate $z$ corresponding to $x…Free
  2. Example 2For the same distribution $X \sim N(50, 100)$, find $P(45 < X < 65)$.Free
  3. Example 3$X$ is normally distributed with mean 200 and standard deviation 5, i.e. $X \sim N(200, 25)$. Find the value of $x$ such that $P(X > x) = 0.…Preview
  4. Example 4The life of an electric bulb manufactured by a company is normally distributed with mean 2,000 hours and standard deviation 150 hours. What…Preview
  5. Example 5Marks obtained by students of a Std-12 Commerce class in a Statistics test are normally distributed with mean 60 and standard deviation 15.…Preview