Q.Following are the marks obtained by 9 students in a mathematics test: 50, 69, 20, 33, 53, 39, 40, 65, 59 The mean deviation from the median is:
(A) 9
(B) 10.5
(C) 12.67
(D) 14.76

The key idea is the Mean Deviation About Median for ungrouped data.

First, arrange the data in ascending order:

$20, 33, 39, 40, 50, 53, 59, 65, 69$

The number of observations is $n=9$. Since $n$ is odd, the median ($M$) is the value of the $\left(\frac{n+1}{2}\right)^{\text{th}}$ observation.

$M = \left(\frac{9+1}{2}\right)^{\text{th}} = 5^{\text{th}}$ observation.

From the sorted data, the $5^{\text{th}}$ observation is 50. So, $M=50$.

Next, calculate the absolute deviations from the median, $|x_i - M|$:

$|20-50|=30$

$|33-50|=17$

$|39-50|=11$

$|40-50|=10$

$|50-50|=0$

$|53-50|=3$

$|59-50|=9$

$|65-50|=15$

$|69-50|=19$ …

When we talk about the "mean deviation," we are essentially trying to quantify the average spread or dispersion of data points around a central value. This central value can be the mean, the median, or even the mode. The choice of the central value significantly impacts the result and its interpretation.

The mean deviation from the median is particularly useful because the sum of absolute deviations is minimized when taken from the median. This property makes it a robust measure of dispersion, less affected by extreme values (outliers) compared to the mean deviation from the mean. It tells us, on average, how far each data point is from the middle value of the dataset.

Here's how to calculate it step-by-step:

1. Order the Data:

   The first step in finding the median is to arrange the data in ascending order. This ensures that the middle value can be correctly identified.

   The given marks are: $50, 69, 20, 33, 53, 39, 40, 65, 59$.

   Arranging them in ascending order:

   $20, 33, 39, 40, 50, 53, 59, 65, 69$

2. Find the Median ($M$):

   The median is the middle value of an ordered dataset. Since we have $n=9$ observations (an odd number), the median is the $\left(\frac{n+1}{2}\right)$-th term.

   Median position $= \frac{9+1}{2} = \frac{10}{2} = 5$-th term.

   Looking at our ordered data, the 5th term is $50$.

   So, the median $M = 50$.

3. Calculate Absolute Deviations from the Median:

   Next, we find the absolute difference between each data point ($x_i$) and the median ($M$). We use absolute values because we are interested in the magnitude of the deviation, not its direction (whether it's above or below the median).

   The absolute deviation for each data point $x_i$ from the median $M$ is given by $|x_i - M|$.

   Let's list these deviations:

   • $|20 - 50| = 30$
   • $|33 - 50| = 17$
   • $|39 - 50| = 11$
   • $|40 - 50| = 10$
   • $|50 - 50| = 0$
   • $|53 - 50| = 3$
   • $|59 - 50| = 9$
   • $|65 - 50| = 15$
   • $|69 - 50| = 19$

   We can present this clearly in a table:

   | $x_i$ | $|x_i - M|$ |

   | :---: | :---------: |

   | 20 | 30 |

   | 33 | 17 |

   | 39 | 11 |

   | 40 | 10 |

   | 50 | 0 |

   | 53 | 3 |

   | 59 | 9 |

   | 65 | 15 | …