Test: Statistics- 1 - JEE MCQ

Arranging the given data in ascending order,
we have 34, 38, 42, 44, 46, 48, 54, 55, 63, 70
Here, Median M = (46+48)/2
=47
(∵ n = 10, median is the mean of 5th and 6th items)
∴ Mean deviation = ∑|xi−M|/n
=∑|xi−47|/10
= (13+9+5+3+1+1+7+8+16+23)/10
=8.6

μ = (22 + 26 + 28 + 20 + 24 + 30)/6
= 150/6 
= 25
x(i) = (xi - μ)²
x(22) = (22-25)² = 9
x(26) = (26-25)² = 1
x(28) = (28-25)² = 9
x(20) = (20-25)² = 25
x(24) = (24-25)² = 1
x(30) = (30-25)² = 25
​
(xi - μ)²  = 70
Standard deviation : [(xi - μ)²]/N
= (70/6)^½
= 3.42

ρ = (b(xy) * b(yx))
But sign of ρρ is same as sign of b(xy), b(yx)
Therefore, ρ = 2/7

Given lines are x+4y=3   (i) 
& 3x+y=15        (ii)
We first check which of the line (i) & (ii) is line of regression y
on x and x on y. Let line x+4y=3 be the line of regression x on y, then
other be the line of regression y on x.
∴ From (i) i.e. line of regression x on y we have x=−4y+3
∴ b(xy)=regression coefficient x on y=−4 and 3x+y=15
∴ y=−3x+15
∴ b(yx)=regression coefficient y on x=-3
Now count r²=b(yx)b(xy)=(−4)(−3)=12
⇒ ∣r∣=2(3)^1/2
​which is not possible as 0≤r²≤1. So our assumption is wrong
and line x+4y=3 is line of regression y on x & 3x+y=15 is line of
regression x on y.
∴ 3x+y=15
⇒ x=(15−y)/3
⇒ x=(15−3)/3   (By putting y=3)
​= 4

Let the other nos. be a and b
then (x+y+1+2+6)/5 = 4.4
x + y = 13 ---------------------(1)
Variance = 8.24


41.2 = 19.88 + (x2 + 19.36 – 8.8x) + (y2 + 19.36 – 8.8y) 
21.32 = x2 + y2 + 38.72 – 8.8(x + y) 
x2 + y2 + 38.72 – 8.8(13) – 21.32 = 0 
(using equation (1)) 
x2 + y2 – 97 = 0 …(2) 
Squaring equation (1) both the sides, 
we get (x + y)2 = (13)^2 
x2 + y2 + 2xy = 169 
97 + 2xy = 169 
(using equation (2)) 
xy = 36 or x = 36/y (1)
⇒ 36/y + y = 13 
y2 + 36 = 13y 
y2 – 13y + 36 = 0 
(y – 4)(y – 9) = 0 
Either (y – 4) = 0 or (y – 9) = 0 
⇒ y = 4 or y = 9 
For y = 4 x = 36/y 
= 36/4 = 12 
For y = 9 
x = 36/9 
x = 4 
Thus, remaining two observations are 4 and 9.

The correlation between x and y is positive correlation because it is a correlation where if one variable increases, the other also increases, and if one variable decreases, the other also decreases.

The relationship between the mean, quartile and the standard deviation are as follows:
Mean Deviation is the mean of all the absolute deviations of a set of data.
Quartile deviation is the difference between “first and third quartiles” in any distribution.
Standard deviation measures the “dispersion of the data set” that is relative to its mean.
Mean Deviation = 4/5 × Quartile deviation
Standard Deviation = 3/2 × Quartile deviation

• Arithmetic Mean (A.M.) : The arithmetic mean is calculated by summing all the observations and dividing by the number of observations. It is highly sensitive to extreme values (outliers) because every observation contributes directly to the sum, and extreme values can significantly skew the result.

• Median : The median is the middle value when the data is arranged in ascending or descending order. It is not affected by extreme values because it only depends on the position of the values, not their magnitudes.

• Mode : The mode is the most frequently occurring value in a dataset. Extreme observations do not affect the mode unless they change the frequency distribution significantly.

• Geometric Mean (G.M.) : The geometric mean is calculated as the nth root of the product of n observations. While it is less sensitive to extreme values compared to the arithmetic mean, it is still more affected than the median or mode.
   

• The arithmetic mean (A.M.) is the most affected by extreme observations.

  Thus, the correct answer is:

  D: A.M.​

To find the mode, we count how many times each number appears:

• 0 appears 3 times.

• 1 appears 1 time.

• 6 appears 5 times.

• 7 appears 2 times.

• 2 appears 2 times.

• 3 appears 1 time.

• 5 appears 1 time.

Since 6 occurs the most frequently (5 times), the mode is 6.

Thus, the correct answer is:

D: 6

The statistical method that allows us to estimate or predict the unknown value of one variable from the known value of a related variable is called regression.

Thus, the correct answer is:

B: regression

Answer: A: |r| ≤ 1

Explanation:

The correlation coefficient (r) measures the strength and direction of the linear relationship between two variables. Its value always lies within the range -1 to +1, inclusive. This means:

• If r = 1, there is a perfect positive linear relationship.

• If r = 0, there is no linear relationship.

• If r = -1, there is a perfect negative linear relationship.

Thus, the absolute value of r (|r|) satisfies |r| ≤ 1. This ensures that the correlation coefficient is always bounded within the interval [-1, 1].

Why other options are incorrect:

• Option B (0 < r < 1): Incorrect because r can also take negative values (for example, -1 < r < 0).

• Option C (|r| > 1): Incorrect because the correlation coefficient cannot exceed 1 in magnitude.

• Option D (-1 < r < 0): Incorrect because r can also take positive values or be 0 or ±1.

Final Answer: A: |r| ≤ 1

Answer: B: 6

Explanation: When a constant is added to every observation in a data set, the spread or dispersion of the data does not change. Therefore, adding 1 to each observation leaves the standard deviation unchanged.

Answer: A: mean

Explanation: The mean is considered the most stable measure of central tendency because it takes into account every value in the data set. This results in it having a lower sampling variability compared to the median or mode, especially as the sample size increases.

Explanation: The total weight of the first group is 10 × 28 = 280. Let the number of items in the second group be n, so its total weight is 35 × n = 35n. For the combined group of (10 + n) items with a mean of 30, the total weight is 30 × (10 + n). Setting up the equation:

280 + 35n = 30 × (10 + n)

Expanding the right side gives:

280 + 35n = 300 + 30n

Subtracting 30n from both sides:

280 + 5n = 300

Subtracting 280 from both sides:

5n = 20

Dividing by 5:

n = 4

Answer: D: quartile deviation

Explanation: Quartile deviation, also known as the semi-interquartile range, is based on the quartiles of the data and is less influenced by extreme values. Unlike measures such as range, standard deviation, or mean deviation, quartile deviation focuses on the middle 50% of the data, making it a more robust measure of dispersion when extreme items are present.

Correlation coefficient = cov (x,y)/ (std deviation (x) ×std deviation (y))
Correlation coefficient  = 0.28
cov (x,y) = 7.6
variance of x is 9.  
=> std deviation (x) = √variance  of X = √9 = 3
=>  0.28  = 7.6 / ( 3 * std deviation (y))
=> std deviation (y) = 7.6 / ( 3 * 0.28)
=> std deviation (y) = 9.05
standard deviation of Y series = 9.05

To calculate the quartile deviation (Q.D.), follow these steps:

1. Arrange the data in ascending order: 7, 10, 12, 15, 17, 19, 25

2. Find the median (middle value). Since there are 7 data points, the median is the 4th value: Median = 15

3. Divide the data into the lower and upper halves (excluding the median):

   • Lower half: 7, 10, 12

   • Upper half: 17, 19, 25

4. Find the first quartile (Q1) which is the median of the lower half: Q1 = 10

5. Find the third quartile (Q3) which is the median of the upper half: Q3 = 19

6. Compute the quartile deviation using: Quartile Deviation = (Q3 - Q1) / 2 = (19 - 10) / 2 = 9 / 2 = 4.5

Thus, the Q.D. of the daily wages is 4.5, and the correct answer is:

A: 4.5

Answer: C: range

Explanation: The mode, mean, and median are measures of central tendency that describe the central point of a data set. The range, on the other hand, measures the spread of the data and is a measure of dispersion, not central tendency.

Answer: A: 75

Explanation: To find the mean, add the numbers and then divide by the count. For the set 130, 126, 68, 50, 1:

1. Calculate the sum: 130 + 126 + 68 + 50 + 1 = 375.

2. Divide the sum by the number of items (5): 375 / 5 = 75.

Median = [mode + 2(mean)]/3
= [6μ+2(9μ)]/3
= 24μ/3
= 8μ