CBSE-CLASS-X-MATH-CHAPTER-13-14-IMPORTANT-QUESTIONS

 CHAPTER 13 STATISTICS & CHAPTER 14 CBSE CLASS 10 MATH IMPORTANT BOARD QUESTIONS

Mean = (Sum of observations) / (Number of observations)

Mean (Discrete data) = (Σfᵢxᵢ) / (Σfᵢ)

Mean (Step Deviation Method) = a + [ (Σfᵢuᵢ) / (Σfᵢ) ] × h
where uᵢ = (xᵢ − a) / h

Mean (Assumed Mean Method) = a + (Σfᵢdᵢ) / (Σfᵢ)
where dᵢ = xᵢ − a

Mean (Grouped data) = (Σfᵢxᵢ) / (Σfᵢ)

Median (Ungrouped data) = value of (n+1)/2 th observation

Median (Grouped data) = l + [ ( (n/2 − cf) / f ) × h ]

Mode (Grouped data) = l + [ ( (f₁ − f₀) / (2f₁ − f₀ − f₂) ) × h ]

Relation between Mean, Median and Mode:
Mode = 3Median − 2Mean

Cumulative Frequency (Less than type)
C.F. = Previous C.F. + frequency

Class Mark (xᵢ) = (Upper limit + Lower limit) / 2

Class Width (h) = Upper limit − Lower limit

Mean = (Σx) / n

Mean (Discrete Frequency) = (Σfᵢxᵢ) / (Σfᵢ)

Mean (Assumed Mean Method) = a + (Σfᵢdᵢ) / (Σfᵢ)
dᵢ = xᵢ − a

Mean (Step Deviation Method) = a + [ (Σfᵢuᵢ) / (Σfᵢ) ] × h
uᵢ = (xᵢ − a) / h

Shortcut: Choose a as class mark of middle class for faster calculation

Median (Ungrouped data) = value of (n+1)/2 th observation

Median (Grouped data) = l + [ (n/2 − cf) / f ] × h

Shortcut: Median class is the class whose cumulative frequency is just greater than n/2

Mode (Grouped data) = l + [ (f₁ − f₀) / (2f₁ − f₀ − f₂) ] × h

Shortcut: Modal class is the class with highest frequency

Relation: Mode = 3Median − 2Mean

Relation: Mean = (3Median − Mode) / 2

Relation: Median = (Mode + 2Mean) / 3

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1. Consider the following distribution of daily wages of 50 workers of a factory.

   Daily Wages (₹)	500–520	520–540	540–560	560–580	580–600
   Number of Workers	12	14	8	6	10

2. The table below shows the daily expenditure on food of 25 households in a locality.

   Daily Expenditure (₹)	100–150	150–200	200–250	250–300	300–350
   Number of Households	4	5	12	2	2

3. The following data gives the information on the observed lifetimes (in hours) of 225 electrical components.

   Lifetimes (hours)	0–20	20–40	40–60	60–80	80–100	100–120
   Frequency	10	35	52	61	38	29

4. A student noted the number of cars passing through a spot on a road for 100 periods, each of 3 minutes, and summarised it in the table below. Find the mode of the data.

   Number of Cars	0–10	10–20	20–30	30–40	40–50	50–60	60–70	70–80
   Frequency	7	14	13	12	20	11	15	8

5. If the median of the distribution given below is 28.5, find the values of x and y.

   Class Interval	Frequency
   0–10	5
   10–20	x
   20–30	20
   30–40	15
   40–50	y
   50–60	5
   Total	60

6. A life insurance agent found the following data for the distribution of ages of 100 policy holders. Calculate the median age, if policies are given only to persons having age 18 years onwards but less than 60 years.

   Age (in years)	Number of Policy Holders
   Below 20	2
   Below 25	6
   Below 30	24
   Below 35	45
   Below 40	78
   Below 45	89
   Below 50	92
   Below 55	98
   Below 60	100

1. A bag contains a red ball, a blue ball and a yellow ball, all the balls being of the same size. Kritika takes out a ball from the bag without looking into it. What is the probability that she takes out:
   (i) a yellow ball?
   (ii) a red ball?
   (iii) a blue ball?
2. One card is drawn from a well-shuffled deck of 52 cards. Find the probability that the card drawn:
   (i) is an ace
   (ii) is not an ace
3. Two players, Sangeeta and Reshma, play a tennis match. It is known that the probability of Sangeeta winning the match is 0.62. What is the probability of Reshma winning the match?
4. A carton consists of 100 shirts of which 88 are good, 8 have minor defects and 4 have major defects. Jimmy, a trader, will only accept shirts which are good, but Sujatha, another trader, will only reject shirts which have major defects. One shirt is drawn at random from the carton. Find the probability that:
   (i) it is acceptable to Jimmy
   (ii) it is acceptable to Sujatha
5. A box contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out at random. Find the probability that the marble taken out is:
   (i) red
   (ii) white
   (iii) not green
6. A piggy bank contains 100 fifty-paise coins, 50 one-rupee coins, 20 two-rupee coins and 10 five-rupee coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down, find the probability that the coin:
   (i) is a 50 paise coin
   (ii) is not a five-rupee coin
7. Twelve defective pens are accidentally mixed with 132 good pens. It is not possible to distinguish between a defective and a good pen by looking at it. One pen is drawn at random. Find the probability that the pen drawn is a good one.
8. (i) A lot of 20 bulbs contains 4 defective bulbs. One bulb is drawn at random from the lot. Find the probability that the bulb drawn is defective.
   
   (ii) Suppose the bulb drawn in part (i) is not defective and is not replaced. Now one bulb is drawn at random from the remaining bulbs. Find the probability that this bulb is not defective.
9. Which of the following arguments are correct and which are not correct? Give reasons for your answer.
   
   (i) If two coins are tossed simultaneously, there are three possible outcomes—two heads, two tails or one of each. Therefore, the probability of each outcome is 1/3.
   
   (ii) If a die is thrown, there are two possible outcomes—an odd number or an even number. Therefore, the probability of getting an odd number is 1/2.

Probability of an Event (E) = n(E) / n(S)

Sample Space (One Coin Toss): S = {H, T}
n(S) = 2

Sample Space (Two Coins Tossed): S = {HH, HT, TH, TT}
n(S) = 4

Sample Space (Three Coins Tossed): S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
n(S) = 8

Sample Space (One Die Thrown): S = {1, 2, 3, 4, 5, 6}
n(S) = 6

Total Cards in a Deck = 52

Number of Red Cards = 26
Number of Black Cards = 26

Number of Cards in Each Suit = 13

Number of Face Cards = 12
(Kings, Queens, Jacks)

Number of Aces = 4

Number of Kings = 4

Number of Queens = 4

Number of Jacks = 4

Probability of Getting a Red Card = 26 / 52 = 1 / 2

Probability of Getting a Black Card = 26 / 52 = 1 / 2

Probability of Getting an Ace = 4 / 52 = 1 / 13

Probability of Getting a Face Card = 12 / 52 = 3 / 13

Complementary Event Formula:
P(Not A) = 1 − P(A)