Any other explanation?

Isnt the median here , the median of two numbers? Or am i wrong?

Explanation:

To find the mean, let us first arrange these 20 terms in ascending order:
464, 505, 564, 585, 604, 608, 616, 653, 657, 660, 697, 704, 741, 856, 888, 906, 918, 926, 947, 1102

Median (for even number of terms) = \(\frac{10^{th} term + 11^{th} term}{2} = \frac{660 + 697}{2} = 678.5\)

Average (given) = 730
SD (given) = 168
1SD of mean means 1SD above or 1SD below mean i.e. (730 - 168) to (730 + 168) = 562 to 898

Find the numbers which are less than the median;
464, 505, 564, 585, 604, 608, 616, 653, 657, 660

Which are not within one standard deviation of the mean;
464, 505, 564, 585, 604, 608, 616, 653, 657, 660

Required probability = \(\frac{2}{20} (100) = 10\)%
Hence, option A

There are several ways to go about this problem, but the objective is to chose the path that will take the least amount of time. Hence, usually you would and certainly can arrange all the numbers from least to greatest to find the median, but I personally believe that this may take too long.

Thus, after reading the problem, the first I did was count how many numbers were given; I counted 10 in the top row, and hence, automatically knew there were a total of 20 because of the alignment with the second row.

Next, if you know the formula for calculating the median, it will make your life much simpler...

There are two formulas used to calculate the median, based upon the number of observations you are given in your data set.

For an odd number of observations (n): (n+1) / 2

For an even numbers of observations (n): [(n / 2) + (n+1) / 2] / 2

In our problem, we have noted that there are 20 numbers, aka 20 observations in our data set.

So, first find which 2 terms you will be using to find their average:

(20 / 2) + (20 + 1) / 2 = 10th term + 11th term

Start crossing out the numbers, starting with 464, until finding your 10th and 11th terms.

10th term = 660
11th term = 697

Average these two numbers, and you get 678.5.

Now, we can start eliminating numbers according to the last sentence of information we are given.

Primarily, we are looking for numbers that are not within 1 standard deviation of the mean.

730 - 168 = 562

So, we are looking for numbers below 562 that are also less than the median, 678.5.

Now, notice that the numbers we have already crossed out are where we are going to find our numbers that fit the criteria.

Finally, we find that the only two numbers that fit this criteria are 464 and 505.

Hence, (2 / 20) x 100% = 10%

And voila, we get our answer!