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Question

# Find the arithmetic mean of each of the following frequency distributions using step-deviation method: Class Number of students 4-8 2 8-12 12 12-16 15 16-20 25 20-24 18 24-28 12 28-32 13 32-36 3

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Solution

## <!-- /* Font Definitions */ @font-face {font-family:"Cambria Math"; panose-1:2 4 5 3 5 4 6 3 2 4; mso-font-charset:0; mso-generic-font-family:roman; mso-font-pitch:variable; mso-font-signature:-536870145 1107305727 0 0 415 0;} @font-face {font-family:Calibri; panose-1:2 15 5 2 2 2 4 3 2 4; mso-font-charset:0; mso-generic-font-family:swiss; mso-font-pitch:variable; mso-font-signature:-536870145 1073786111 1 0 415 0;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-unhide:no; mso-style-qformat:yes; mso-style-parent:""; margin-top:0cm; margin-right:0cm; margin-bottom:10.0pt; margin-left:0cm; line-height:115&#37;; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:Calibri; mso-fareast-theme-font:minor-latin; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi; mso-fareast-language:EN-US;} span.SpellE {mso-style-name:""; mso-spl-e:yes;} .MsoChpDefault {mso-style-type:export-only; mso-default-props:yes; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:Calibri; mso-fareast-theme-font:minor-latin; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi; mso-fareast-language:EN-US;} .MsoPapDefault {mso-style-type:export-only; margin-bottom:10.0pt; line-height:115&#37;;} @page WordSection1 {size:595.3pt 841.9pt; margin:72.0pt 72.0pt 72.0pt 72.0pt; mso-header-margin:35.4pt; mso-footer-margin:35.4pt; mso-paper-source:0;} div.WordSection1 {page:WordSection1;} --> Class Frequency $\left({f}_{i}\right)$ Mid values$\left({x}_{i}\right)$ ${u}_{i}=\frac{\left({x}_{i}-A\right)}{h}$ $\text{=}\frac{\left({x}_{i}-18\right)}{4}$ $\left({f}_{i}×{u}_{i}\right)$ 4-8 2 6 -3 -6 8-12 12 10 -2 -24 12-16 15 14 -1 -15 16-20 25 18=A 0 0 20-24 18 22 1 18 24-28 12 26 2 24 28-32 13 30 3 39 32-36 3 34 4 12 $\sum {f}_{i}=100$ $\sum \left({f}_{i}×{u}_{i}\right)=48$ $\text{Now,}A=18,h=4,\sum {f}_{i}=100\mathrm{and}\sum \left({f}_{{}_{i}}×{u}_{i}\right)=48\phantom{\rule{0ex}{0ex}}\therefore \text{Mean,}\overline{x}=A+\left\{h×\frac{\sum \left({f}_{{}_{i}}×{u}_{i}\right)}{\sum {f}_{i}}\right\}\phantom{\rule{0ex}{0ex}}\text{=18+}\left\{4×\frac{48}{100}\right\}\phantom{\rule{0ex}{0ex}}\text{=18+1}\text{.92}\phantom{\rule{0ex}{0ex}}\text{=19}\text{.92}\phantom{\rule{0ex}{0ex}}\therefore \overline{x}=19.92$

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