Question

Uniformity of irrigation is checked in a border strip by measuring the water penetration depth at 16 equally spaced stations along the strip. The depths of penetration of water recorded (in mm) are as follows:

0.65	0.83	0.79	0.87
0.67	0.85	0.86	0.68
0.88	0.77	0.73	0.59
0.89	0.63	0.72	0.70

The distribution uniformity low-quarter (DULQ), in fraction, is ________.

Hint:

Sort the readings, isolate the lowest quarter of stations, then compare that group's average to the overall average.

Answer 1

Correct Answer: 0.84

Step 1: Arrange the readings in increasing order.
The 16 recorded depths (mm) are: 0.65, 0.83, 0.79, 0.87, 0.67, 0.85, 0.86, 0.68, 0.88, 0.77, 0.73, 0.59, 0.89, 0.63, 0.72, 0.70.
Sorting them from lowest to highest gives: 0.59, 0.63, 0.65, 0.67, 0.68, 0.70, 0.72, 0.73, 0.77, 0.79, 0.83, 0.85, 0.86, 0.87, 0.88, 0.89.

Step 2: Pick out the lowest quarter.
"Low-quarter" means the lowest one-fourth of the readings. With 16 stations, that is the lowest \( 16/4 = 4 \) values.
The lowest 4 readings are 0.59, 0.63, 0.65, and 0.67 mm.

Step 3: Average the lowest quarter.
\[ \bar{d}_{lq} = \frac{0.59 + 0.63 + 0.65 + 0.67}{4} = \frac{2.54}{4} = 0.635 \text{ mm} \]

Step 4: Average all 16 readings.
Adding all 16 values gives a total of 12.11 mm, so:
\[ \bar{d} = \frac{12.11}{16} = 0.756875 \text{ mm} \]

Step 5: Apply the DULQ formula.
Distribution uniformity low-quarter compares the average of the driest quarter to the overall average:
\[ DU_{LQ} = \frac{\bar{d}_{lq}}{\bar{d}} = \frac{0.635}{0.756875} = 0.839 \]

Final Answer:
Rounded to two decimal places, the distribution uniformity low-quarter is 0.84, which lies inside the official range of 0.81 to 0.88. \[ \boxed{DU_{LQ} = 0.84} \]