Exercise 2.2 gave the following data (in increasing order) for the attribute age: 13, 15, 16, 16, 19, 20, 20, 21, 22, 22, 25, 25

Question

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Exercise 2.2 gave the following data (in increasing order) for the attribute age: 13, 15, 16, 16, 19, 20, 20, 21, 22, 22, 25, 25

, 25, 25, 30, 33, 33, 35, 35, 35, 35, 36, 40, 45, 46, 52, 70. (a) Use smoothing by bin means to smooth these data, using a bin depth of 3. Illustrate your steps. Comment on the effect of this technique for the given data. (b) How might you determine outliers in the data? (c) What other methods are there for data smoothing?

Mathematics

1 answer:

babunello [11.8K] · Answer 1 · 2026-02-16T05:15:09+03:00

Answer:

Step-by-step explanation:

Data given in ascending order: 13, 15, 16, 16, 19, 20, 20, 21, 22, 22, 25, 25, 25, 25, 30, 33, 33, 35, 35, 35, 35, 36, 40, 45, 46, 52, 70.

a) Smoothing via bin means:

Bins of size 3:

Bin 1: 13, 15, 16

Bin 2: 16, 19, 20

Bin 3: 20, 21, 22

Bin 4: 22, 25, 25

Bin 5: 25, 25, 30

Bin 6: 33, 33, 35

Bin 7: 35, 35, 35

Bin 8: 36, 40, 45

Bin 9: 46, 52, 70

Calculate mean for each bin and replace each value with this mean:

Bin 1 mean: (13+15+16)/3 = 15; replaced with 15,15,15

Bin 2 mean: (16+19+20)/3 = 18; replaced with 18,18,18

Bin 3 mean: (20+21+22)/3 = 21; replaced with 21,21,21

Bin 4 mean: (22+25+25)/3 = 24; replaced with 24,24,24

Bin 5 mean: (25+25+30)/3 = 27; replaced with 27,27,27

Bin 6 mean: (33+33+35)/3 = 34; replaced with 34,34,34

Bin 7 mean: (35+35+35)/3 = 35; replaced with 35,35,35

Bin 8 mean: (36+40+45)/3 = 40; replaced with 40,40,40

Bin 9 mean: (46+52+70)/3 = 56; replaced with 56,56,56

This method reduces noise in the data, helping produce more reliable results.

b) Outlier identification:

Outliers are values significantly different from the rest. One may detect them using box-and-whisker plots (box plots) or by calculating the interquartile range.

c) Other smoothing techniques include:

- Binning by boundaries

- Exponential smoothing

- Random walk smoothing