Question

The number of times each week that a factory machine broke down was noted over a period of 50 consecutive weeks. The results are given in the following table:

Number of breakdowns	0	1	2	3	4	5	6
Number of weeks	2	12	14	8	8	4	2

A. Find the mean number of breakdown in this period. Is this value exact or an estimate?
B. Give the mode and median of the number of breakdowns.

Answer 1

Hint: We will first write down the data given in the question and then we will find the mean by applying the formula that is $\overline{x}=\dfrac{\sum{{{f}_{i}}{{x}_{i}}}}{\sum{{{x}_{i}}}}$ and then for second part we will see that the median will be ${{\left( \dfrac{n+1}{2} \right)}^{th}}$term and mode will be the most occurring value and in this way we will find the answer.

Complete step by step answer:
First, let’s consider the first part, that is we have to find the number of breakdown in this given period.
We have:
Let $x$ be the number of breakdowns, $f$ be the frequency of $x$ occurring therefore number of weeks will be the frequency

Number of weeks $\left( {{f}_{i}} \right)$	Number of breakdowns $\left( {{x}_{i}} \right)$	${{f}_{i}}{{x}_{i}}$
2	0	0
12	1	12
14	2	28
8	3	24
8	4	32
4	5	20
2	6	12

Now, from the above data: $\sum{{{f}_{i}}=50}$ and $\sum{{{f}_{i}}{{x}_{i}}=128}$
Now, we know that the sample mean will be: $\overline{x}=\dfrac{\sum{{{f}_{i}}{{x}_{i}}}}{\sum{{{x}_{i}}}}$
Now, we will put the values from our data: $\overline{x}=\dfrac{\sum{{{f}_{i}}{{x}_{i}}}}{\sum{{{x}_{i}}}}=\dfrac{128}{50}=2.56$
Therefore the mean value for the number of breakdowns will be 2.56.
Now this is an estimated value as the breakdowns should be a whole number and the mean values are in decimals therefore the mean calculated will be an estimated value.
Now, we will move on to the second part of the question and find the median and the mode of the number of the number of breakdowns.
Now, median is the middle value of the given set of data when arranged in a particular order. Now, since our number of observations in the given data is odd, therefore the median will be ${{\left( \dfrac{n+1}{2} \right)}^{th}}$ observation that is ${{\left( \dfrac{7+1}{2} \right)}^{th}}={{4}^{th}}$ observation, therefore the median will be = 3
Now, the mode of a data is the most frequent data occurring in the given set, therefore we will have our mode as: 2 as it occurs for a maximum number of time that is for 14 weeks.

Hence, Mean is 2.56 , Median is 3 and Mode will be 2

Note: When finding median remember that when the number of observations is even then the median will be both ${{\left( \dfrac{n}{2} \right)}^{th}}\text{ and }{{\left( \dfrac{n}{2}+1 \right)}^{th}}$ observation. In these types of questions be careful while writing down the data as students may make silly mistakes while writing down the data.