Math 132A

Numerical Summaries

Quartiles

  • First quartile

    • Separates the lowest quarter of the values from the highest three quarters.

  • Second quartile

    • Separates the lowest half of the values from the highest half.

  • Third quartile

    • Separates the lowest three quarters of the values from the highest quarter.

Five numbers summary:

  • Minimum

  • First quartile

  • Second quartile (median)

  • Third quartile

  • Maximum

Five numbers summary:

1 1 2 4 6 7 7 9 10 10 10 13 14 15 16 20
Min Q1 Median Q3 Max
1 5 9.5 13.5 20

Mean

1 1 2 4 6 7 7 9 10 10 10 13 14 15 16 20

Standard Deviation

1 1 2 4 6 7 7 9 10 10 10 13 14 15 16 20

 

12345678910111213141516171819209.062513−9.06256−9.062520−9.06251−9.0625deviation:𝑥−𝑥=13−9.0625=3.9375deviation:𝑥−𝑥=6−9.0625=−3.0625deviation:𝑥−𝑥=20−9.0625=10.9375deviation:𝑥−𝑥=1−9.0625=−8.0625variance=”mean”ofsquaresofallthedeviationsvariance=1𝑛−1∑(𝑥−𝑥)2standarddeviation:𝑠=√variance

\(x\) \(x - \overline{x}\) \((x - \overline{x})^2\)
1 -8.0625 65.0039
1 -8.0625 65.0039
2 -7.0625 49.8789
4 -5.0625 25.6289
6 -3.0625 9.3789
7 -2.0625 4.2539
7 -2.0625 4.2539
9 -0.0625 0.0039
10 0.9375 0.8789
10 0.9375 0.8789
10 0.9375 0.8789
13 3.9375 15.5039
14 4.9375 24.3789
15 5.9375 35.2539
16 6.9375 48.1289
20 10.9375 119.6289
145 468.9374

\(\displaystyle\overline{x} = \frac{145}{16} = 9.0625\)

  • Sum of square deviations: \[\sum (x - \overline{x})^2 = 468.9374\]
  • Variance: \[\frac{1}{15}\sum (x - \overline{x})^2 = \frac{468.9374}{15} = 31.2625\]
  • Standard Deviation: \[\sqrt{\frac{1}{15}\sum (x - \overline{x})^2} = \sqrt{31.2625} = 5.5912879\]

Summary

  • Measures of center:

    • Median
    • Mean

  • Measures of variation (spread)

    • Range
    • IQR
    • Variance
    • Standard deviation