STATISTICS QUESTIONS (with detailed solutions)

Statistics Questions

Question 1

(a) Find the mean deviation from the median for the following data:

x	15	21	27	30	35
f	3	5	6	7	8

(b) A random sample of 120 bean seeds were collected, each was weighed to the nearest 0.01gm and the results were summarized below:

Weight (gm)	1.10 - 1.29	1.30 - 1.49	1.50 - 1.69	1.70 - 1.89	1.90 - 2.09	2.10 - 2.29	2.30 - 2.49
Number of seeds	7	24	33	32	14	6	4

Calculate:

i. The quartile deviation

ii. The standard deviation using coding method with A = 1.795 correct to 4 decimal places

Question 2

(a) An incomplete frequency distribution table is given below:

X	10-19	20-29	30-39	40-49	50-59	60-69	70-79	Total

Given that the median is 46. Find:

i. The value of the missing frequency F₁ and F₂

ii. The mean and variance using coding method (take the assumed mean in the class of 40-49)

Question 3

The following table shows the size of shoes sold with the respective number of pairs:

Size	5	5.5	6	6.5	7	7.5	8	8.5	9	9.5
No. of pairs	2	5	16	30	60	40	23	11	4	1

Compute:

i. The first and third quartile

ii. Interquartile range

iii. Range

Question 4

The mean and standard deviation of 20 observations are found to be 10 and 2 respectively. On rechecking it was found that an observation 8 was incorrect. If the wrong item is replaced by 12:

Calculate the new mean and standard deviation

Question 5

Calculate the mean, variance and standard deviation for this distribution of the test results of 200 students:

Marks	0-20	20-30	30-40	40-50	50-60	60-70	70-80	80-90

Statistics Questions with Detailed Solutions

Question 1

(a) Find the mean deviation from the median for the following data:

x	15	21	27	30	35
f	3	5	6	7	8

Solution:

Step 1: Calculate cumulative frequencies

x	f	Cumulative f
15	3	3
21	5	8
27	6	14
30	7	21
35	8	29

Step 2: Find the median
Total frequency N = 29
Median position = (N+1)/2 = (29+1)/2 = 15th item
From cumulative frequency, 15th item falls in x = 27

Step 3: Calculate absolute deviations from median

x	f	|x - Median|	f × |x - Median|
15	3	|15-27| = 12	36
21	5	|21-27| = 6	30
27	6	|27-27| = 0	0
30	7	|30-27| = 3	21
35	8	|35-27| = 8	64
Total			151

Step 4: Calculate mean deviation
Mean Deviation = (Σf|x - Median|) / N = 151 / 29 ≈ 5.2069

(b) Bean seeds weight distribution:

Weight (gm)	1.10-1.29	1.30-1.49	1.50-1.69	1.70-1.89	1.90-2.09	2.10-2.29	2.30-2.49
Number of seeds	7	24	33	32	14	6	4

Solution:

Part i: Quartile Deviation

Step 1: Calculate cumulative frequencies

Class	f	Cumulative f
1.10-1.29	7	7
1.30-1.49	24	31
1.50-1.69	33	64
1.70-1.89	32	96
1.90-2.09	14	110
2.10-2.29	6	116
2.30-2.49	4	120

Step 2: Find Q1 (First Quartile)
Q1 position = N/4 = 120/4 = 30th item
Falls in class 1.50-1.69
Q1 = L + [(N/4 - CF)/f] × h
Where:
L = lower boundary = 1.495
CF = cumulative frequency before = 31
f = frequency of class = 33
h = class width = 0.2
Q1 = 1.495 + [(30-31)/33] × 0.2 ≈ 1.489

Step 3: Find Q3 (Third Quartile)
Q3 position = 3N/4 = 90th item
Falls in class 1.70-1.89
Q3 = L + [(3N/4 - CF)/f] × h
L = 1.695, CF = 64, f = 32, h = 0.2
Q3 = 1.695 + [(90-64)/32] × 0.2 ≈ 1.8575

Step 4: Calculate Quartile Deviation
QD = (Q3 - Q1)/2 = (1.8575 - 1.489)/2 ≈ 0.18425

Part ii: Standard Deviation using Coding Method (A = 1.795)

Step 1: Create coding table

Class	Midpoint (x)	f	u = (x-A)/h	fu	fu²
1.10-1.29	1.195	7	-3	-21	63
1.30-1.49	1.395	24	-2	-48	96
1.50-1.69	1.595	33	-1	-33	33
1.70-1.89	1.795	32	0	0	0
1.90-2.09	1.995	14	1	14	14
2.10-2.29	2.195	6	2	12	24
2.30-2.49	2.395	4	3	12	36
Total				-64	266

Step 2: Calculate mean of coded data
Mean (ū) = Σfu/N = -64/120 ≈ -0.5333

Step 3: Calculate variance of coded data
Variance (u) = [Σfu²/N] - (Σfu/N)² = [266/120] - (-0.5333)² ≈ 2.2167 - 0.2844 = 1.9323

Step 4: Convert to original scale
h = class width = 0.2
Variance (x) = h² × Variance (u) = 0.04 × 1.9323 ≈ 0.07729
Standard Deviation = √0.07729 ≈ 0.2780

Question 2

(a) Incomplete frequency distribution table with median = 46

Solution:

Part i: Find missing frequencies F₁ and F₂

Assumption: The table has classes 10-19, 20-29, ..., 70-79 with two missing frequencies F₁ and F₂

Step 1: Let total frequency = N
N = Sum of all frequencies = F₁ + F₂ + other known frequencies

Step 2: Median position
Median is 46, which falls in class 40-49
Median formula: L + [(N/2 - CF)/f] × h = 46
Where:
L = 39.5 (lower boundary)
CF = cumulative frequency before median class = F₁ + F₂ + sum of previous classes
f = frequency of median class
h = class width = 10

Step 3: Need more information
Note: The complete table with all frequencies is needed to solve this precisely.
Typically, we would establish two equations from:
1. Total frequency
2. Median condition
And solve simultaneously for F₁ and F₂

Part ii: Mean and Variance using Coding Method

Step 1: Choose assumed mean A = midpoint of 40-49 = 44.5

Step 2: Create coding table
u = (x - A)/h where h = class width = 10

Step 3: Calculate mean
Mean = A + (Σfu/N) × h

Step 4: Calculate variance
Variance = [Σfu²/N - (Σfu/N)²] × h²

Note: Complete frequency distribution is needed for exact calculations.

Question 3

Shoe size distribution:

Size	5	5.5	6	6.5	7	7.5	8	8.5	9	9.5
No. of pairs	2	5	16	30	60	40	23	11	4	1

Solution:

Part i: First and Third Quartile

Step 1: Calculate cumulative frequencies

Size	f	Cumulative f
5	2	2
5.5	5	7
6	16	23
6.5	30	53
7	60	113
7.5	40	153
8	23	176
8.5	11	187
9	4	191
9.5	1	192

Step 2: Find Q1 (First Quartile)
Q1 position = N/4 = 192/4 = 48th item
Falls at size 6.5

Step 3: Find Q3 (Third Quartile)
Q3 position = 3N/4 = 144th item
Falls at size 7.5

Part ii: Interquartile Range
IQR = Q3 - Q1 = 7.5 - 6.5 = 1.0

Part iii: Range
Range = Maximum - Minimum = 9.5 - 5 = 4.5

Question 4

Original data: mean = 10, standard deviation = 2, n = 20

Incorrect observation: 8 replaced with 12

Solution:

Step 1: Calculate original sum
Original mean = 10 = Σx/n ⇒ Σx = 10 × 20 = 200

Step 2: Calculate new sum
New sum = Original sum - incorrect value + correct value
= 200 - 8 + 12 = 204

Step 3: Calculate new mean
New mean = New sum / n = 204 / 20 = 10.2

Step 4: Calculate original sum of squares
Original variance = 2² = 4 = (Σx²/n) - mean² ⇒ Σx² = (4 + 100) × 20 = 2080

Step 5: Calculate new sum of squares
New Σx² = Original Σx² - incorrect value² + correct value²
= 2080 - 64 + 144 = 2160

Step 6: Calculate new variance
New variance = (Σx²/n) - new mean² = (2160/20) - 10.2² = 108 - 104.04 = 3.96

Step 7: Calculate new standard deviation
New SD = √3.96 ≈ 1.99

Question 5

Test results of 200 students:

Marks	0-20	20-30	30-40	40-50	50-60	60-70	70-80	80-90

Solution:

Assumption: Frequencies are missing in the question. For demonstration, we'll assume arbitrary frequencies.

General Approach:

Step 1: Find midpoints of each class
Example: 0-20 → 10, 20-30 → 25, etc.

Step 2: Choose assumed mean A (usually middle class midpoint)

Step 3: Calculate coded values u = (x - A)/h
Where h = class width (varies by class)

Step 4: Calculate mean
Mean = A + (Σfu/N) × h

Step 5: Calculate variance
Variance = [Σfu²/N - (Σfu/N)²] × h²

Step 6: Standard deviation = √Variance

Note: Exact calculations require frequency distribution for each class interval.