HOW TO COMPUTE QUARTILES, DECILES AND PERCENTILES FOR UNGROUPED DATA

Main Ideas

Quantiles are values that divide an ordered distribution into equal-sized parts. They can also help describe a distribution’s shape. For example, if the upper quartile is farther from the median than the lower quartile, the distribution may be skewed to the right.

The video covers three types of quantiles:

• Quartiles divide data into four parts. The first, second, and third quartiles correspond to the 25th, 50th, and 75th percentiles.
• Deciles divide data into ten parts. For example, the sixth decile corresponds to 60%.
• Percentiles divide data into 100 parts.

Method for Ungrouped Data

1. Order the observations from smallest to largest and count them to find (n).

2. Find the position of the desired quantile using: [ r = \frac{k}{m}n + \left(1 - \frac{k}{m}\right) ] where (m=4) for quartiles, (m=10) for deciles, and (m=100) for percentiles. The value (k) identifies the requested quantile.

3. Read or interpolate the value:

   • If the position is a whole number, use the observation at that position.
   • If the position falls between two observations, interpolate. For a position (j+d), use: [ x_j + d(x_{j+1}-x_j) ]

Worked Example

For the ordered data 6, 8, 10, 12, 12, 14, 15, 16, 20, there are (n=9) observations.

• Third quartile (Q_3): The position is 7, so (Q_3=15). This means 75% of the scores are at or below 15.

• Sixth decile (D_6): The position is 5.8. Interpolating between the fifth value (12) and the sixth value (14) gives: [ 12 + 0.8(14-12) = 13.6 ] This represents the 60th percentile.

• 65th percentile (P_{65}): The position is 6.2. Interpolating between the sixth value (14) and the seventh value (15) gives: [ 14 + 0.2(15-14) = 14.2 ] Thus, approximately 65% of the scores are at or below 14.2.

Speaker

The video features one speaker: the narrator or instructor. No other speakers or sources are identified.