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 letters | 1 - 4 | 4 - 7 | 7 - 10 | 10 - 13 | 13 - 16 | 16 - 19 |
---|---|---|---|---|---|---|
Number of surnames | 6 | 30 | 40 | 16 | 4 | 4 |
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.
The cumulative frequencies with their respective class intervals are as follows.
Number of letters | Frequency (f\(_i\)) | Cumulative frequency |
---|---|---|
1 - 4 | 6 | 6 |
4 - 7 | 30 | 30 + 6 = 36 |
7 - 10 | 40 | 40 + 36 = 76 |
10 - 13 | 16 | 76 + 16 = 92 |
13 - 16 | 4 | 92 + 4 = 96 |
16 - 19 | 4 | 96 + 4 = 100 |
Total (n) | 100 |
|
Cumulative frequency just greater \(\frac{n}2 ( i.e., \frac{100}2 = 50)\) than is 76, belonging to class interval 7−10.
Median class = 7−10
Lower limit (\(l\)) of median class = 7
Frequency (\(f\)) of median class = 36
Cumulative frequency (\(cf\)) of median class = 40
Class size (\(h\)) = 3
Median = \(l + (\frac{\frac{n}2 - cf}f \times h)\)
Median = \(7 + (\frac{50 - 36}{40} \times 3)\)
Median = 7 + \(\frac{40 \times 3}{40}\)
Median = 8.05
To find the class mark (xi) for each interval, the following relation is used.
Class mark \((x_i)\) = \(\frac {\text{Upper \,limit + Lower \,limit}}{2}\)
Taking 11.5 as assumed mean (a), \(d_i\), \(u_i\), and \(f_iu_i\) are calculated according to step deviation method as follows.
Number of letters | Frequency (f\(_i\)) | \(\bf{x_i}\) | \(\bf{d_i = x_i -11.5}\) | \(\bf{u_i = \frac{d_i}{3}}\) | \(\bf{f_iu_i}\) |
---|---|---|---|---|---|
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 | 100 |
|
|
| -106 |
From the table, it can be observed that
\(\sum f_i = 100\)
\(\sum f_iu_i = -106\)
Mean, \(\overset{-}{x} = a + (\frac{\sum f_iu_i}{\sum f_i})\times h\)
\(\overset{-}{x}\) = \(11.5 + (\frac{-106 }{100})\times 3\)
\(\overset{-}{x}\) = 11.5 - 3.18
Mean, \(\overset{-}{x}\) = 8.32
The data in the given table can be written as
Number of letters | Frequency (f\(_i\)) |
---|---|
1 - 4 | 6 |
4 - 7 | 30 |
7 - 10 | 40 |
10 - 13 | 16 |
13 - 16 | 4 |
16 - 19 | 4 |
Total | 100 |
From the data given above, it can be observed that the maximum class frequency is 40, belonging to class interval 7 - 10.
Therefore, modal class = 7 - 10
Lower limit (\(l\)) of modal class = 7
Frequency (\(f_1\)) of modal class = 40
Frequency (\(f_0\)) of class preceding the modal class = 30
Frequency (\(f_2\)) of class succeeding the modal class = 16
Class size (\(h\)) = 3
Mode = \(l\) + \((\frac{f_1 - f_0 }{2f_1 - f_0 - f_2)} \times h\)
Mode = \(7 + (\frac{40 - 30 }{ 2(40) - 30 - 16}) \times3\)
Mode =\(7+ [\frac{10}{34}] \times 3\)
Mode = \(7 +( \frac{ 30}{ 34})\)
Mode = 7 + 0.88
Mode = 7.88
Therefore, median number and mean number of letters in surnames is 8.05 and 8.32 respectively while modal size of surnames is 7.88.
The following data shows the number of family members living in different bungalows of a locality:
Number of Members | 0−2 | 2−4 | 4−6 | 6−8 | 8−10 | Total |
---|---|---|---|---|---|---|
Number of Bungalows | 10 | p | 60 | q | 5 | 120 |
If the median number of members is found to be 5, find the values of p and q.
The population of lions was noted in different regions across the world in the following table:
Number of lions | Number of regions |
---|---|
0–100 | 2 |
100–200 | 5 |
200–300 | 9 |
300–400 | 12 |
400–500 | x |
500–600 | 20 |
600–700 | 15 |
700–800 | 10 |
800–900 | y |
900–1000 | 2 |
Total | 100 |
If the median of the given data is 525, find the values of x and y.
प्रादेशिक स्तर पर आयोजित होने वाली 100 मीटर की बाधा दौड़ में आपके मित्र को प्रथम स्थान मिला है। उसे बधाई देते हुए लगभग 40 शब्दों में एक संदेश लिखिए।
‘मेघला के आकार वाली पर्वत श्रृंखला ने पृथ्वी को चारों तरफ से घेर रखा है।’ – रेखांकित पदों की जगह उपयुक्त समस्तपद प्रस्तुत कीजिए तथा समास का नाम भी लिखिए।