Find the best tutors and institutes for Class 10 Tuition
Q6:
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:
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.
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:
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.
Solution :
Given: A frequency distribution of the number of letters in 100 surnames.
| Number of letters | Number of surnames ($f_i$) |
|---|---|
| 1 - 4 | 6 |
| 4 - 7 | 30 |
| 7 - 10 | 40 |
| 10 - 13 | 16 |
| 13 - 16 | 4 |
| 16 - 19 | 4 |
To Find: The Median, Mean, and Mode of the given data.
Step 1: Calculation of Mean ($\bar{x}$)
We use the Assumed Mean Method. Let the assumed mean $a = 11.5$. The class size $h = 3$.
| Class Interval | Frequency ($f_i$) | Class Mark ($x_i$) | $d_i = x_i - a$ | $u_i = \frac{x_i - a}{h}$ | $f_i u_i$ |
|---|---|---|---|---|---|
| 1-4 | 6 | 2.5 | -9 | -3 | -18 |
| 4-7 | 30 | 5.5 | -6 | -2 | -60 |
| 7-10 | 40 | 8.5 | -3 | -1 | -40 |
| 10-13 | 16 | 11.5 | 0 | 0 | 0 |
| 13-16 | 4 | 14.5 | 3 | 1 | 4 |
| 16-19 | 4 | 17.5 | 6 | 2 | 8 |
| Total | $\sum f_i = 100$ | - | - | - | $\sum f_i u_i = -106$ |
Formula for Mean: $\bar{x} = a + \left( \frac{\sum f_i u_i}{\sum f_i} \right) \times h$
$\bar{x} = 11.5 + \left( \frac{-106}{100} \right) \times 3 = 11.5 - 3.18 = 8.32$
Step 2: Calculation of Median
Total frequency $N = 100$. So, $N/2 = 50$.
| Class Interval | Frequency ($f_i$) | Cumulative Frequency ($cf$) |
|---|---|---|
| 1-4 | 6 | 6 |
| 4-7 | 30 | 36 |
| 7-10 | 40 | 76 |
The cumulative frequency just greater than 50 is 76, so the median class is $7-10$.
Lower limit ($l$) = 7, $cf$ of preceding class = 36, $f$ of median class = 40, $h = 3$.
Median = $l + \left( \frac{N/2 - cf}{f} \right) \times h = 7 + \left( \frac{50 - 36}{40} \right) \times 3 = 7 + \left( \frac{14}{40} \right) \times 3 = 7 + 1.05 = 8.05$
Step 3: Calculation of Mode
The modal class is the class with the highest frequency, which is $7-10$ ($f_1 = 40$).
$l = 7, f_1 = 40, f_0 = 30, f_2 = 16, h = 3$.
Mode = $l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h = 7 + \left( \frac{40 - 30}{2(40) - 30 - 16} \right) \times 3$
Mode = $7 + \left( \frac{10}{80 - 46} \right) \times 3 = 7 + \left( \frac{10}{34} \right) \times 3 = 7 + 0.88 = 7.88$
Final Answer: The Mean number of letters is 8.32, the Median is 8.05, and the Mode is 7.88.
More Questions from Class 10 Mathematics Statistics EXERCISE 13.3
- Q1: 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.
- Q2: If the median of the distribution given below is 28.5, find the values of $x$ and $y$.
- Q3: 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.
- Q4: 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 : 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.)
- Q5: The following table gives the distribution of the life time of 400 neon lamps : Find the median life time of a lamp.
- Q7: The distribution below gives the weights of 30 students of a class. Find the median weight of the students.
CBSE Solutions for Class 10 Mathematics Statistics
Chapters in CBSE - Class 10 Mathematics
Top Tutors who teach Statistics
Top Tutor
Alamganj, Patna
4 yrs of
Exp
9 students
16 hrs of classes completed
200
per hour
Student's Home
Tutor's home
I am teaching in school ( Dalimss Sunbeam school Ghazipur ) as a PGT teacher.. I have over all 4 years experience in this field.
Top Tutor
Mohali SAS Nagar, Chandigarh
15 yrs of
Exp
18 students
397 hrs of classes completed
500
per hour
Online Classes
Tutor's home
I have more than 15 years of experience in teaching maths and science for class 10 .. I also prepare students for various competition exams like olympiads, NDA etc
She teaches better than every teacher she does conceptual clarity and gives more homework for practice and she helps me to give occasional tests to improve my score
Top Tutor
Munirka, Delhi
3 yrs of
Exp
1 student
48 hrs of classes completed
200
per hour
Online Classes
I am a teacher since last 3-4 years. I have teaching tuition to students of class 9th and 10th CBSE board. In these covid times i prefer online classes and my subjects of expertise are science, maths ans social science.
Top Tutor
Godrej Garden City, Ahmedabad
6 yrs of
Exp
20 students
1554 hrs of classes completed
500
per hour
Online Classes
I’m a Teacher having 5years experience and offering Home/Online tution of Mathematics & Science for CBSE/ GSEB Students of class 8 to 12. I’m Bachelor of Engineer. My teaching / coaching / mentoring style is such that the students can start and accelerate from whatever their current level. I am sure with my right efforts an average students also will be transformed into best performer.
His classes were very useful as they helped me get good marks in cbse 10th, maths. He is very disciplined and approachable.
Top Tutor
Kamal Chowk, Nagpur
20 yrs of
Exp
Tutor's home
I have 20 years of experience in the field of teaching. I conduct classes for Spoken English, BBA, MBA, Hindi Language, Career Counselling and Classes I to X. I tutor (offline/online) for syllabi-based subjects Classes I to X for CBSE, ICSE, IGCSE, IB Board. For classes X and above, I focus on tutoring subjects like Hindi, English, Psychology. Accountancy, Business Studies, Mathematics, Physics, Chemistry for CBSE, ICSE, IGCSE and IB Board. I also conduct career counselling and psychological counselling. I also assist students in pursuing MBA/PGDBM courses in their dissertation work. People engaged in doctoral theses in the domains of management and psychology engage my services for their research work. Since there is a lot of demand for enhancing one's conversational English, this is what I am focusing on at the moment. Being bilingual, my medium of communication is English as well as Hindi. I train learners in the usage of WordPress, Joomla - to develop their websites; right from domain registration to site maintenance. Besides, I assist teachers in pursuing Teachers' Training Courses.
Find more Tutor for Statistics in your City
- Bangalore Mathematics Tutors
- Delhi Mathematics Tutors
- Chennai Mathematics Tutors
- Gurgaon Mathematics Tutors
- Noida Mathematics Tutors
- Hyderabad Mathematics Tutors
- Mumbai Mathematics Tutors
- Ghaziabad Mathematics Tutors
- Chandigarh Mathematics Tutors
- Pune Mathematics Tutors
- Jaipur Mathematics Tutors
- Surat Mathematics Tutors
