Looking for complete, step-by-step textbook solutions for SEBA Class 10 Mathematics Chapter 13 (Statistics - Exercise 13.3) for the 2026–27 academic session? Access verified, board-pattern solutions to all textual questions on finding the Median of grouped data using cumulative frequency distribution tables (cf). Master the high-yield median formula (Median = l + [((n/2) - cf) / f] * h), solving the famous 5-mark missing frequencies x and y problem with median 28.5, converting "less than" cumulative data into continuous class intervals (life insurance policies), calculating lengths of 40 leaves with continuous boundary adjustments, comparing mean, median, and mode for empirical verification (3 Median = Mode + 2 Mean), and exam-oriented model solutions designed strictly according to the Assam Board (SEBA) HSLC blueprint.

1. The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the median, mean and mode of the data and compare them.

Monthly consumption (in units)Number of consumers
65 – 854
85 – 1055
105 – 12513
125 – 14520
145 – 16514
165 – 1858
185 – 2054

Soln:

Monthly consumption (in units)Number of consumers (fi​)Cumulative freq. (cf)Class mark (xi​)ui​=20xi​−135​fi​ui​
65 – 854475-3-12
85 – 1055995-2-10
105 – 1251322115-1-13
125 – 1452042135 (a)00
145 – 1651456155114
165 – 185864175216
185 – 205468195312
Totalsum f_i = 68———

(i) Median:

n = 68 ⇒ n/2 = 68 / 2 = 34

The cumulative frequency just greater than 34 is 42, which corresponds to the class interval 125 – 145.

Median class = 125 – 145

Here:

l = 125, n/2 = 34, cf = 22, f = 20, h = 20

Formula:

Median = l + [((n/2) – cf) / f] × h

Substitution:

Median = 125 + [(34 – 22) / 20] × 20

Median = 125 + 12

Median = 137 units

(ii) Mean: Assumed mean (a) = 135 Class size (h) = 20

Formula:

Mean (x̄) = a + (Σfᵢuᵢ / Σfᵢ) × h

Substitution:

x̄ = 135 + (7 / 68) × 20

x̄ = 135 + 140 / 68

x̄ = 135 + 2.06

x̄ = 137.06 units

(iii) Mode: The maximum frequency is 20, corresponding to the class interval 125 – 145.

Modal class = 125 – 145

Here:

l = 125, f₁ = 20, f₀ = 13, f₂ = 14, h = 20

Formula:

Mode = l + [(f₁ – f₀) / (2f₁ – f₀ – f₂)] × h

Substitution:

Mode = 125 + [(20 – 13) / (2(20) – 13 – 14)] × 20

Mode = 125 + [7 / (40 – 27)] × 20

Mode = 125 + (140 / 13)

Mode = 125 + 10.77

Mode = 135.77 units

Comparison:

Median = 137 units

Mean = 137.06 units

Mode = 135.77 units

All three measures of central tendency are approximately equal in this data.

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

Class intervalFrequency
0 – 105
10 – 20x
20 – 3020
30 – 4015
40 – 50y
50 – 605
Total60

Soln: Given:

Median = 28.5

Total frequency (n) = 60

Class intervalFrequency (fi​)Cumulative frequency (cf)
0 – 1055
10 – 20x5 + x
20 – 302025 + x
30 – 401540 + x
40 – 50y40 + x + y
50 – 60545 + x + y
Totaln = 60—

From the table:

45 + x + y = 60

x + y = 60 – 45

x + y = 15 …(1)

Since Median = 28.5, it lies in the class interval 20 – 30. Median class = 20 – 30

Here:

l = 20, n/2 = 60 / 2 = 30, cf = 5 + x, f = 20, h = 10

Formula:

Median = l + [((n/2) – cf) / f] × h

Substitution:

28.5 = 20 + [{30 – (5 + x)} / 20] × 10

28.5 – 20 = (25 – x) / 2

8.5 = (25 – x) / 2

17 = 25 – x

x = 25 – 17

x = 8

Substituting x = 8 in equation (1):

8 + y = 15

y = 15 – 8

y = 7

Ans: x = 8, y = 7

3. A life insurance agent found the following data for 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 year.

Age (in years)Number of policy holders
Below 202
Below 256
Below 3024
Below 3545
Below 4078
Below 4589
Below 5092
Below 5598
Below 60100

Soln: Converting the given cumulative frequency distribution into class intervals:

Age (in years)Number of policy holders (cf)Frequency (fi​)
18 – 2022
20 – 2566 – 2 = 4
25 – 302424 – 6 = 18
30 – 354545 – 24 = 21
35 – 407878 – 45 = 33
40 – 458989 – 78 = 11
45 – 509292 – 89 = 3
50 – 559898 – 92 = 6
55 – 60100100 – 98 = 2
Total—n = 100

n = 100 ⇒ n/2 = 100 / 2 = 50

The cumulative frequency just greater than 50 is 78, which corresponds to the class interval 35 – 40.

Median class = 35 – 40

Here:

l = 35, n/2 = 50, cf = 45, f = 33, h = 5

Formula:

Median = l + [((n/2) – cf) / f] × h

Substitution:

Median = 35 + [(50 – 45) / 33] × 5

Median = 35 + (5 / 33) × 5

Median = 35 + 25 / 33

Median = 35 + 0.76

Median = 35.76 years

SEBA Class 10 Maths Exercise 13.3 Solutions

4. The lengths of 40 leaves of a plant are measured correct to the nearest millimetre, and the data obtained is represented in the following table:

Length (in mm)Number of leaves
118 – 1263
127 – 1355
136 – 1449
145 – 15312
154 – 1625
163 – 1714
172 – 1802

Find the median length of the leaves. (Hint: The data needs to be converted to continuous classes for finding the median, since the formula assumes continuous classes. The classes then change to 117.5 – 126.5, 126.5 – 135.5, …, 171.5 – 180.5.)

Soln: Converting the given discontinuous classes into continuous classes by subtracting 0.5 from lower limits and adding 0.5 to upper limits:

Length (in mm)Number of leaves (fi​)Cumulative frequency (cf)
117.5 – 126.533
126.5 – 135.558
135.5 – 144.5917
144.5 – 153.51229
153.5 – 162.5534
162.5 – 171.5438
171.5 – 180.5240
Totaln = 40—

n = 40 ⇒ n/2 = 40 / 2 = 20

The cumulative frequency just greater than 20 is 29, which corresponds to the class interval 144.5 – 153.5.

Median class = 144.5 – 153.5

Here:

l = 144.5, n/2 = 20, cf = 17, f = 12, h = 9

Formula:

Median = l + [((n/2) – cf) / f] × h

Substitution:

Median = 144.5 + [(20 – 17) / 12] × 9

Median = 144.5 + (3 / 12) × 9

Median = 144.5 + (1 / 4) × 9

Median = 144.5 + 2.25

Median = 146.75 mm

Ans: Median length of the leaves = 146.75 mm

5. The following table gives the distribution of the life time of 400 neon lamps:

Life time (in hours)Number of lamps
1500 – 200014
2000 – 250056
2500 – 300060
3000 – 350086
3500 – 400074
4000 – 450062
4500 – 500048

Find the median life time of a lamp.

Soln:

Life time (in hours)Number of lamps (fi​)Cumulative frequency (cf)
1500 – 20001414
2000 – 25005670
2500 – 300060130
3000 – 350086216
3500 – 400074290
4000 – 450062352
4500 – 500048400
Totaln = 400—

n = 400 ⇒ n/2 = 400 / 2 = 200

The cumulative frequency just greater than 200 is 216, which corresponds to the class interval 3000 – 3500.

Median class = 3000 – 3500

Here:

l = 3000, n/2 = 200, cf = 130, f = 86, h = 500

Formula:

Median = l + [((n/2) – cf) / f] × h

Substitution:

Median = 3000 + [(200 – 130) / 86] × 500

Median = 3000 + (70 / 86) × 500

Median = 3000 + 35000 / 86

Median = 3000 + 406.98

Median = 3406.98 hours

Ans: Median life time of a lamp = 3406.98 hours

6. 100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabets in the surnames was obtained as follows:

Number of lettersNumber of surnames
1 – 46
4 – 730
7 – 1040
10 – 1316
13 – 164
16 – 194

Determine the median number of letters in the surnames. Find the mean number of letters in the surnames? Also, find the modal size of the surnames.

Soln: Assumed mean (a) = 8.5

Class size (h) = 3

uᵢ = (xᵢ – 8.5) / 3

Number of lettersfi​cfxi​ui​=3xi​−8.5​fi​ui​
1 – 4662.5-2-12
4 – 730365.5-1-30
7 – 1040768.5 (a)00
10 – 13169211.5116
13 – 1649614.528
16 – 19410017.5312
Totalsum f_i = 100———

(i) Median:

n = 100 ⇒ n/2 = 50

Median class = 7 – 10 l = 7, cf = 36, f = 40, h = 3

Median = l + [((n/2) – cf) / f] × h

Median = 7 + [(50 – 36) / 40] × 3

Median = 7 + (14 / 40) × 3

Median = 7 + 42 / 40

Median = 7 + 1.05

Median = 8.05

(ii) Mean:

Mean (x̄) = a + (Σfᵢuᵢ / Σfᵢ) × h

x̄ = 8.5 + (-6 / 100) × 3

x̄ = 8.5 – 18 / 100

x̄ = 8.5 – 0.18

x̄ = 8.32

(iii) Mode:

Modal class = 7 – 10 (maximum frequency = 40) l = 7, f₁ = 40, f₀ = 30, f₂ = 16, h = 3

Mode = l + [(f₁ – f₀) / (2f₁ – f₀ – f₂)] × h

Mode = 7 + [(40 – 30) / (2(40) – 30 – 16)] × 3

Mode = 7 + [10 / (80 – 46)] × 3

Mode = 7 + 30 / 34

Mode = 7 + 0.88

Mode = 7.88

Ans:

Median = 8.05

Mean = 8.32

Mode = 7.88

Class 10 Maths Chapter 13.3 Statistics SEBA 2026-27

7. The distribution below gives the weights of 30 students of a class. Find the median weight of the students.

Weight (in kg)Number of students
40 – 452
45 – 503
50 – 558
55 – 606
60 – 656
65 – 703
70 – 752

Soln:

Weight (in kg)Number of students (fi​)Cumulative frequency (cf)
40 – 4522
45 – 5035
50 – 55813
55 – 60619
60 – 65625
65 – 70328
70 – 75230
Totaln = 30—

n = 30 ⇒ n/2 = 30 / 2 = 15

The cumulative frequency just greater than 15 is 19, which corresponds to the class interval 55 – 60.

Median class = 55 – 60

Here:

l = 55, n/2 = 15, cf = 13, f = 6, h = 5

Formula:

Median = l + [((n/2) – cf) / f] × h

Substitution:

Median = 55 + [(15 – 13) / 6] × 5

Median = 55 + (2 / 6) × 5

Median = 55 + 5 / 3

Median = 55 + 1.67

Median = 56.67 kg

Ans: Median weight = 56.67 kg

8. Determine the value of median for the following data

MarksNo. of Students
Less than 105
Less than 2015
Less than 3098
Less than 40242
Less than 50367
Less than 60405
Less than 70425
Less than 80438
Less than 90439

Soln:

MarksCumulative frequency (cf)Class IntervalFrequency (fi​)
Less than 1050 – 105
Less than 201510 – 2015 – 5 = 10
Less than 309820 – 3098 – 15 = 83
Less than 4024230 – 40242 – 98 = 144
Less than 5036740 – 50367 – 242 = 125
Less than 6040550 – 60405 – 367 = 38
Less than 7042560 – 70425 – 405 = 20
Less than 8043870 – 80438 – 425 = 13
Less than 9043980 – 90439 – 438 = 1
Totaln = 439——

n = 439 ⇒ n/2 = 439 / 2 = 219.5

The cumulative frequency just greater than 219.5 is 242, which corresponds to the class interval 30 – 40.

Median class = 30 – 40

Here:

l = 30, n/2 = 219.5, cf = 98, f = 144, h = 10

Formula:

Median = l + [((n/2) – cf) / f] × h

Substitution:

Median = 30 + [(219.5 – 98) / 144] × 10

Median = 30 + (121.5 / 144) × 10

Median = 30 + 1215 / 144

Median = 30 + 8.44

Median = 38.44

Ans: Median = 38.44

9. During the medical check-up of 35 students of a class, their weights were recorded as follows:

Weight (in kg)No. of Students
Less than 380
Less than 403
Less than 425
Less than 449
Less than 4614
Less than 4828
Less than 5032
Less than 5235

Calculate the median weight from the data

Soln:

Weight (in kg)Cumulative frequency (cf)Class IntervalFrequency (fi​)
Less than 380Below 380
Less than 40338 – 403 – 0 = 3
Less than 42540 – 425 – 3 = 2
Less than 44942 – 449 – 5 = 4
Less than 461444 – 4614 – 9 = 5
Less than 482846 – 4828 – 14 = 14
Less than 503248 – 5032 – 28 = 4
Less than 523550 – 5235 – 32 = 3
Totaln = 35——

n = 35 ⇒ n/2 = 35 / 2 = 17.5

The cumulative frequency just greater than 17.5 is 28, which corresponds to the class interval 46 – 48.

Median class = 46 – 48

Here:

l = 46, n/2 = 17.5, cf = 14, f = 14, h = 2

Formula:

Median = l + [((n/2) – cf) / f] × h

Substitution:

Median = 46 + [(17.5 – 14) / 14] × 2

Median = 46 + (3.5 / 14) × 2

Median = 46 + 7 / 14

Median = 46 + 0.5

Median = 46.5 kg

Ans: Median weight = 46.5 kg

SEBA Class 10 Maths Ex 13.3 Question Answer

10. Which of the following statements about statistics are true?

(i) Statistics involves data collection

(ii) Statistics helps in drawing conclusions

(iii) All data must be quantitative

(iv) Statistics can be misleading if misused

Choose correct option

(a) (i) and (ii) are true

(b) (i), (ii) and (iv) are true

(c) (iii) and (iv) are true

(d) (i), (ii) and (iii) are true

Soln: Analysis of statements:

(i) Statistics involves data collection: True

(ii) Statistics helps in drawing conclusions: True

(iii) All data must be quantitative: False (data can also be qualitative)

(iv) Statistics can be misleading if misused: True

Hence, statements (i), (ii), and (iv) are true.

Correct Option: (b)

11. Below is the data for weekly earnings (in Rs.) of workers in a factory.

Earnings (in Rs.)No. of workers
5000-60006
6000-700010
7000-800022
8000-900010
9000-100002

What is the cumulative frequency up to income Rs. 9000?

(a) 16

(b) 38

(c) 48

(d) 50

Soln:

Earnings (in Rs.)No. of workers (fi​)Cumulative frequency (cf)
5000 – 600066
6000 – 7000106 + 10 = 16
7000 – 80002216 + 22 = 38
8000 – 90001038 + 10 = 48
9000 – 10000248 + 2 = 50
Totaln = 50—

Cumulative frequency up to income Rs. 9000 corresponds to the upper limit of the class 8000 – 9000, which is 48.

Correct Option: (c)

Ans: (c) 48

12. Match the formulas with the measure given in list (I) and list (II), then choose the correct option from the given alternatives

List-(I)List-(II)
P. Σfx / Σf(i) Median
Q. l + [((n/2) – c.f.) / f] × h(ii) Mode
R. l + [(f₁ – f₀) / (2f₁ – f₀ – f₂)] × h(iii) Mean
S. Highest value – Lowest value(iv) Range

(a) P → (iii), Q → (i), R → (ii), S → (iv)

(b) P → (i), Q → (ii), R → (iii), S → (iv)

(c) P → (iii), Q → (ii), R → (iv), S → (i)

(d) P → (iii), Q → (iv), R → (i), S → (ii)

Soln: Matching:

Order: P → (iii), Q → (i), R → (ii), S → (iv)

Correct Option: (a)

SEBA HSLC Class 10 Maths Chapter 13 Exercise 13.3 Solved PDF

13. The following table shows the marks obtained by 60 students in a mathematics test

Class interval (Marks)Number of Students (frequency)
0-102
10-204
20-3010
30-4011
40-5015
50-6010
60-703
70-802
80-902
90-1001

Which is the median class of the above frequency table?

(a) 40-50

(b) 50-60

(c) 60-70

(d) 30-40

Soln:

Class interval (Marks)Frequency (fi​)Cumulative frequency (cf)
0 – 1022
10 – 2046
20 – 301016
30 – 401127
40 – 501542
50 – 601052
60 – 70355
70 – 80257
80 – 90259
90 – 100160
Totaln = 60—

n = 60 ⇒ n/2 = 60 / 2 = 30

The cumulative frequency just greater than 30 is 42, which corresponds to the class interval 40 – 50.

Median class = 40 – 50

Correct Option: (a)

14. If the mean is 20 and the median is 18, then the mode is. (a) 18 (b) 22 (c) 14 (d) 26

Soln: Given:

Mean = 20

Median = 18

Empirical Formula:

Mode = 3(Median) – 2(Mean)

Substitution:

Mode = 3(18) – 2(20)

Mode = 54 – 40

Mode = 14

Correct Option: (c)

15. In annual sports meet of a school, a 100 m race was organized. A table shown below describes the time in which the race was completed by the number of students

Time (in sec)0-2020-4040-6060-8080-100 PNG
No. of students6131038

Based on the above data answer the following questions:

(i) The lower limit of the modal class is

(a) 30

(b) 40

(c) 60

(d) 20

(ii) The average time taken by the student to finish the race is

(a) 40

(b) 47

(c) 50

(d) 30

(iii) The cumulative frequency table is constucted to determine

(a) Mean

(b) Mode

(c) Median

(d) All of above

(iv) How many students finished the race within 1 minute?

(a) 30

(b) 40

(c) 19

(d) 29

Soln:

Time (in sec)No. of students (fi​)Class mark (xi​)fi​xi​cf
0 – 20610606
20 – 40133039019
40 – 60105050029
60 – 8037021032
80 – 10089072040
TotalSum of fi = 40—Sum of (fi * xi) = 1880—

(i) Lower limit of the modal class:

Highest frequency is 13, corresponding to the modal class 20 – 40.

Lower limit = 20

Correct Option: (d)

(ii) Average time taken to finish the race:

Mean (x̄) = Σfᵢxᵢ / Σfᵢ

x̄ = 1880 / 40

= 47 seconds

Correct Option: (b)

(iii) Purpose of cumulative frequency table:

Cumulative frequency table is constructed to determine the Median.

Correct Option: (c)

(iv) Number of students who finished within 1 minute (≤ 60 seconds):

Students with time 0 – 60 seconds = 6 + 13 + 10 = 29

Correct Option: (d)

📊 HSLC Class 10 Maths Exercise 13.3 Notice: This page features the complete, step-by-step SEBA Class 10 Mathematics Chapter 13 (Statistics) Exercise 13.3 Official Textual Solutions updated for the 2026–27 academic session. Formatted according to the strict HSLC board marking scheme, every numerical problem—from setting up cumulative frequency tables(cf), locating the median class using n/2, to working through the high-weightage missing frequencies x and y (where total frequency is 60 and median is 28.5)—is worked out with clear step-by-step algebraic simplification to ensure you secure full marks in your 4-mark and 5-mark board exam questions.

💡 Master SEBA Class 10 Median & Statistics:

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