Question

A manufacturing company finds out the number of defective calculators in each of the last $10$ batches of $100$ calculators.


Which measure of central tendency is least impacted with or without the outlier?

• A. Mean
• B. Median
• C. Mode
• D. None of the above

Answer 1

The correct option is C Mode

From the given data set, we observe that in the batch number $10$, the number of defective calculators is $98$ out of $100$ calculators. As something goes terribly wrong in this batch number $10$, we can consider $98$ defective calculators as an outlier here.

$\underset{–}{\text{Calculation of Mean:}}$

We know,
$\text{Mean of a data set}=\frac{\text{Sum of all data points}}{\text{Total number of data points}}$

$\bullet$ Including outlier:

The mean number of defective calculators with outlier is

$\frac{23+34+43+13+45+27+38+42+16+98}{10}$
$=\frac{379}{10}=37.9$

$\bullet$ Excluding outlier:

The mean number of defective calculators without outlier is

$\frac{23+34+43+13+45+27+38+42+16}{9}$
$=\frac{281}{9}=31.2\overline{2}$

So, we get the mean number of defective calculators
$\bullet$ with outlier= $37.9$
$\bullet$ without outlier= $31.2\overline{2}$

$\therefore$ The outlier impacts the mean of the given data set.

$\underset{–}{\text{Calculation of Median:}}$

For calculating the median, we need to first arrange the number of defective calculators in differebnt batches in an increasing order as shown below.

$13$

$16$

$23$

$27$

$34$

$38$

$42$

$43$

$45$

$98$

$\bullet$ Including outlier:

The median of the data set including outlier= mean of $5^{\text{th}}\&6^{\text{th}}$ data values

$=\frac{34+38}{2}=\frac{72}{2}=36$

$\bullet$ Excluding outlier:

Excluding outlier, the data set can be represented as

$13$

$16$

$23$

$27$

$\underset{–}{34}$

$38$

$42$

$43$

$45$

The median of the data set excluding outlier is $5^{\text{th}}$ data value, i.e., $34$.

So, we get the median number of defective calculators
$\bullet$ with outlier= $36$
$\bullet$ without outlier= $34$

$\therefore$ The outlier also impacts the median of the given data set.

$\underset{–}{\text{Calculation of Mode:}}$

$\bullet$ Including outlier:

The given data set including outlier can be represented as

$13$

$16$

$23$

$27$

$34$

$38$

$42$

$43$

$45$

$98$

Here, the number of defective calculators in all $10$ batches are different.

The mode of the data set including outlier= No mode

$\bullet$ Excluding outlier:

The given data set excluding outlier can be represented as

$13$

$16$

$23$

$27$

$34$

$38$

$42$

$43$

$45$

Here, the number of defective calculators in all $9$ batches are different.

The mode of the data set excluding outlier= No mode

So, we get the mode number of defective calculators
$\bullet$ with outlier= No mode
$\bullet$ without outlier= No mode

$\therefore$ The outlier does not impact the mode of the given data set.

Hence, here, mode is the least impacted central tendency with or without outlier.