{"id":12705,"date":"2023-01-07T20:57:01","date_gmt":"2023-01-07T15:27:01","guid":{"rendered":"https:\/\/mathemerize.com\/?p=12705"},"modified":"2023-01-07T20:57:05","modified_gmt":"2023-01-07T15:27:05","slug":"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-surname-was-obtained-as-follows","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/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-surname-was-obtained-as-follows\/","title":{"rendered":"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 surname was obtained as follows :"},"content":{"rendered":"\n

Question :<\/h2>\n\n\n\n
Number of Letters<\/td>1 – 4<\/td>4 – 7<\/td>7 – 10<\/td>10 – 13<\/td>13 – 16<\/td>16 – 19<\/td><\/tr>
Number of Surnames<\/td>6<\/td>30<\/td>40<\/td>16<\/td>4<\/td>4<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

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.<\/p>\n\n\n\n

Solution :<\/h2>\n\n\n\n

Calculation of median<\/strong> :<\/p>\n\n\n\n

First, we prepare the following table to compute the median :<\/p>\n\n\n\n

Number of Letter<\/td>Number of surnames (frequency)<\/td>cumulative frequency<\/td><\/tr>
1 – 4<\/td>6<\/td>6<\/td><\/tr>
4 – 7<\/td>30<\/td>36<\/td><\/tr>
7 – 10<\/td>40<\/td>76<\/td><\/tr>
10 – 13<\/td>16<\/td>92<\/td><\/tr>
13 – 16<\/td>4<\/td>96<\/td><\/tr>
16 – 19<\/td>4<\/td>100<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

We have : n = 100, so \\(n\\over 2\\) = 50<\/p>\n\n\n\n

The cumulative frequency just greater than 50 is 76 and the corresponding class is (7 – 10). Thus, (7 – 10) is the median class such that<\/p>\n\n\n\n

\\(n\\over 2\\) = 50, l = 7, cf = 36, f = 40 and h = 3.<\/p>\n\n\n\n

Substituting these values in the formula,<\/p>\n\n\n\n

Median = l + (\\({n\\over 2} – cf\\over f\\))\\(\\times \\) h<\/p>\n\n\n\n

= 7 + (\\(50 – 26\\over 40\\))(3) = 7 + 1.05 = 8.05<\/strong><\/p>\n\n\n\n

Calculation of mean<\/strong> :<\/p>\n\n\n\n

Number of letters<\/td>Mid-value (\\(x_i\\))<\/td>Frequency (\\(f_i\\))<\/td>\\(f_ix_i\\)<\/td><\/tr>
1 – 4<\/td>2.5<\/td>6<\/td>15<\/td><\/tr>
4 – 7<\/td>5.5<\/td>30<\/td>165<\/td><\/tr>
7 – 10<\/td>8.5<\/td>40<\/td>340<\/td><\/tr>
10 – 13<\/td>11.5<\/td>16<\/td>184<\/td><\/tr>
13 – 16<\/td>14.5<\/td>4<\/td>58<\/td><\/tr>
16 – 19<\/td>17.5<\/td>4<\/td>70<\/td><\/tr>
Total<\/td><\/td>100<\/td>832<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Mean = \\(\\sum f_ix_i\\over \\sum f_i\\) = \\(832\\over 100\\) = 8.32<\/strong><\/p>\n\n\n\n

Calculation of mode<\/strong> :<\/p>\n\n\n\n

The class (7 – 10) has the maximum frequency. Therefore, this is the modal class.<\/p>\n\n\n\n

Here, l = 7, h = 3, \\(f_1\\) = 40, \\(f_0\\) = 30 and \\(f_2\\) = 16<\/p>\n\n\n\n

Now, let us substitute these values in the formula<\/p>\n\n\n\n

Mode = l + (\\(f_1 – f_0\\over 2f_1 – f_0 – f_2\\))(h) = 7 + \\(10\\over 34\\) \\(\\times\\) 3 = 7 + 0.88 = 7.88<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"

Question : 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 …<\/p>\n

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 surname was obtained as follows :<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[43,58],"tags":[],"yoast_head":"\n100 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 surname was obtained as follows : - Mathemerize<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/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-surname-was-obtained-as-follows\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"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 surname was obtained as follows : - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"Question : 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. 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