Which of the following has the same mean, median and mode?

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MCQ

Which of the following has the same mean, median and mode$?$

A
$6, 2, 5, 4, 3, 4, 1$
B
$4, 2, 2, 1, 3, 2, 3$
C
$2, 3, 7, 3, 8, 3, 2$
✓
$4, 3, 4, 3, 4, 6, 4$

✓

Answer

Correct option: D.

$4, 3, 4, 3, 4, 6, 4$

$(a).$ Data $($in ascending order$) \rightarrow 1, 2, 3, 4, 4, 5, 6$

Here, $n = 7 ($odd$)$
Median $=$ Value of $\Big(\frac{\text{n+1}}{2}\Big)^\text{th}$ observation $=$ Value of $\Big(\frac{8}{2}\Big)^\text{th}$ observation $= 4$
$\text{Mean}=\frac{\text{Sum of observation}}{\text{n}}$
$=\frac{1+2+3+4+4+5+6}{7}$
$=\frac{25}{7}$
$=3.57$
Mode $=$ Most frequent observation $= 4$
Hence,
Mean $\neq$ Median $=$ Mode
$(b).$ Data $($in ascending order$) \rightarrow 1, 2, 2, 2, 3, 3, 4$
Here, $n = 7 ($odd$)$
Median $=$ Value of $\Big(\frac{\text{n+1}}{2}\Big)^\text{th}$ observation $=$ Value of $\Big(\frac{7+1}{2}\Big)^\text{th}$ observation $= 2$
$\text{Mean}=\frac{\text{Sum of observation}}{\text{n}}$
$=\frac{1+2+2+2+3+3+4}{7}$
$=\frac{17}{7}$
$=2.428$
Mode $=$ Most frequent observation $= 2$
Hence,
Mean $\neq$ Median $=$ Mode
$(c).$ Data $($in ascending order$) \rightarrow 2, 2, 3, 3, 3, 7, 8$
Here, $n = 7 ($odd$)$
Median $=$ Value of $\Big(\frac{\text{n+1}}{2}\Big)^\text{th}$ observation $=$ Value of $\Big(\frac{7+1}{2}\Big)^\text{th}$ observation $= 3$
$\text{Mean}=\frac{\text{Sum of observation}}{\text{n}}$
$=\frac{2+2+3+3+3+7+8}{7}$
$=\frac{28}{7}$
$=4$
Hence,
Mean $\neq$ Median $=$ Mode
$(d).$ Data $($in ascending order$) \rightarrow 3, 3, 4, 4, 4, 4, 6$
Here, $n = 7 ($odd$)$
Median $=$ Value of $\Big(\frac{\text{n+1}}{2}\Big)^\text{th}$ observation $=$ Value of $\Big(\frac{7+1}{2}\Big)^\text{th}$ observation $= 4$
$\text{Mean}=\frac{\text{Sum of observation}}{\text{n}}$
$=\frac{3+3+4+4+4+4+6}{7}$
$=\frac{28}{7}$
$=4$
Hence,
Mean $=$ Mode $=$ Median

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