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Range

The range is defined to be the highest value minus the lowest value. The symbol R is used for the range.
range didefinisikan sebagai nilai tertinggi dikurangi nilai terendah. Simbol R digunakan untuk range.
R = highest value – lowest value.
Extremely large or extremely small data values can drastically affect the range.
Nilai data yang sangat besar atau sangat kecil dapat mempengaruhi range secara drastis.

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Population Variance

The variance is the average of the squares of the distance each value is from the mean.
The symbol for the population variance is σ2\sigma^2 (σ\sigma is the Greek lowercase letter sigma)
σ2=Σ(X−μ)2N\sigma^2 = \dfrac{\Sigma (X-\mu)^2}{N}, where
  • XX = is individual value
  • μ\mu = is population mean
  • NN = is population size

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Example

Consider the following data to constitute the population: 10, 60, 50, 30, 40, 20.
Find the mean and variance.
The mean μ\mu = (10 + 60 + 50 + 30 + 40 + 20)/6 = 210/6 = 35.
The variance σ2=Σ(X−μ)2N=17506\sigma^2 = \dfrac{\Sigma (X-\mu)^2}{N} = \dfrac{1750}{6} = 291.67.
Table
XXX−μX-\mu(X−μ)2(X-\mu)^2
10-25625
6025625
5015225
30-525
40525
20-15225
2101750

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Population Standard Deviation

The standard deviation, denoted by σ\sigma, The standard deviation is the square root of the variance.
σ=σ2=Σ(X−μ)2N\sigma = \sqrt{\sigma^2} = \sqrt{\dfrac{\Sigma (X-\mu)^2}{N}}

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Sample Variance

The unbiased estimator of the population variance or the sample variance is a statistic whose value approximates the expected value of a population variance.
It is denoted by s2s^2, where
s2=Σ(X−X‾)2n−1s^2 = \dfrac{\Sigma (X-\overline{\rm X})^2}{n-1}, and
  • X‾\overline{\rm X} = sample mean
  • nn = sample size

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Example

Find the variance and standard deviation for the following sample: 16, 19, 15, 15, 14.
ΣX\Sigma X = (16 + 19 + 15 + 15 + 14) = 79
ΣX2\Sigma X^2 = (16^2 + 19^2 + 15^2 + 15^2 + 14^2) = 1263
s2=ΣX2−(ΣX)2/nn−1s^2 = \dfrac{\Sigma X^2 - (\Sigma X)^2 / n}{n-1}
    =1263−(79)2/54=3.7~~~~ = \dfrac{1263 - (79)^2 / 5}{4} = 3.7
s=3.7=1.9s = \sqrt{3.7} = 1.9

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Sample Standard Deviation

The sample standard deviation is the square root of the sample variance.
s=s2=Σ(X−X‾)2n−1s = \sqrt{s^2} = \sqrt{\dfrac{\Sigma (X-\overline{\rm X})^2}{n-1}}
Shortcut Formula for the Sample Variance and the Standard Deviation
s2=ΣX2−(ΣX)2/nn−1s^2 = \dfrac{\Sigma X^2 - (\Sigma X)^2 / n}{n-1}, or
s=ΣX2−(ΣX)2/nn−1s = \sqrt{\dfrac{\Sigma X^2 - (\Sigma X)^2 / n}{n-1}}

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Sample Variance for Grouped and Ungrouped Data

For grouped data, use the class midpoints for the observed value in the different classes. For ungrouped data, use the same formula with the class midpoints, XmX_m, replaced with the actual observed X value.

The sample variance for grouped data:
s2=Σf∙Xm2−[(Σf∙Xm)2/n]n−1s^2 = \dfrac{\Sigma f \bullet X_m^2 - [(\Sigma f \bullet X_m)^2 / n]}{n-1}
For ungrouped data, replace XmX_m with the observe XX value.

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Sample Variance for Grouped Data - Example

XXfff∙Xf \bullet Xf∙X2f \bullet X^2
521050
6318108
7856392
81864
9654486
10440400
n=24n = 24f∙X=186f \bullet X = 186f∙X2=1500f \bullet X^2 = 1500

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Sample Variance for Ungrouped Data - Example

The sample variance and standard deviation:
s2=Σf∙X2−[(Σf∙X)2/n]n−1s^2 = \dfrac{\Sigma f \bullet X^2 - [(\Sigma f \bullet X)^2 / n]}{n-1}
    =1500−[(186)2/24]23=2.54~~~~ = \dfrac{1500 - [(186)^2 / 24]}{23} = 2.54
s=2.54=1.6s = \sqrt{2.54} = 1.6

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Coefficient of Variation

The coefficient of variation is defined to be the standard deviation divided by the mean. The result is expressed as a percentage.
Koefisien variasi didefinisikan sebagai standar deviasi dibagi dengan rata-rata. Hasilnya dinyatakan dalam persentase.

CVar=sX‾×100CVar = \dfrac{s}{\overline{\rm X}} \times 100 or CVar=σμ×100CVar = \dfrac{\sigma}{\mu} \times 100

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Chebyshev’s Theorem

The proportion of values from a data set that will fall within k standard deviations of the mean will be at least 1−1/k21 - 1/k^2, where k is any number greater than 1.
Proporsi nilai dari kumpulan data yang akan jatuh dalam k standar deviasi dari rata-rata setidaknya 1−1/k21 - 1/k^2, di mana k adalah angka lebih besar dari 1.
For k = 2, 75% of the values will lie within 2 standard deviations of the mean. For k = 3, approximately 89% will lie within 3 standard deviations.
Untuk k = 2, 75% nilai akan berada dalam 2 standar deviasi dari rata-rata. Untuk k = 3, sekitar 89% akan berada dalam 3 standar deviasi.

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The Empirical (Normal) Rule

For any bell shaped distribution:
Untuk distribusi berbentuk lonceng:
  • Approximately 68% of the data values will fall within one standard deviation of the mean.
    Sekitar 68% nilai data akan berada dalam satu standar deviasi dari rata-rata.
  • Approximately 95% will fall within two standard deviations of the mean.
    Sekitar 95% akan berada dalam dua standar deviasi dari rata-rata.
  • Approximately 99.7% will fall within three standard deviations of the mean.
    Sekitar 99.7% akan berada dalam tiga standar deviasi dari rata-rata.
Empirical Rule