Calculating Mode and Median for Non-decreasing Series | Stats Homework Solution

[jaus tail]

Homework Statement

Consider non-decreasing series of numbers: 1, 8, 8, 13, 14, 14, x, y, 18, 20, 31, 34, 38 and 40.
Median is 15
Calculate mode

Homework Equations

Median is middle term for odd number of terms and it's average of middle 2 terms for even number of terms.
3 Median = Mode + 2 Mean

The Attempt at a Solution

Given Median is 5
And 1, 8, 8, 13, 14, 14, x, y, 18, 20, 31, 34, 38 and 40---> this is even number of terms. 14 terms. Median is (x + y)/2
Which is 15
So x + y = 15 * 2 = 30
So we can calculate mean as Summation of numbers = 1 + 8 + 8 + 13 + 14 + 14 + x + y+ 18 + 20 + 31 + 38 + 40
which is 235
Divide by 14 gives mean which is 16.79
So 3 Median = Mode + 2 Mean.
This gives:
3 * 15 = Mode + 2 *16.79
Mode = 11.42

But book gives options:
14
16
18
Can't be determined and this is given to be correct answer.

 

[andrewkirk]

3 Median = Mode + 2 Mean

That formula is incorrect. Where did you get it?

A counter-example is the sample 1, 1, 2
mode = 1
median = 1
mean = 4/3
3 * median = 3
Mode + 2 * mean = 3 + 2/3

There cannot be any equality relationship between mean, median and mode, since in some samples we can change the mean without altering the median or mode.

 

[jaus tail]

Book says about empirical formula.
For moderately symmetrical distribution: mode = 3 median + 2 mean
For symmetrical distribution: mode = mean = median.
Even google says above formula for empirical .
http://math.tutorvista.com/statistics/mean-median-and-mode.html

 

[Ray Vickson]

Book says about empirical formula.
For moderately symmetrical distribution: mode = 3 median + 2 mean
For symmetrical distribution: mode = mean = median.
Even google says above formula for empirical .
http://math.tutorvista.com/statistics/mean-median-and-mode.html

Nevertheless, that is generally wrong. Your data sample has TWO modes: 8 and 14. I can easily devise data samples where the mode and the figure [3 median + 2 mean] can differ by hundreds.