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z score

The standard score or z score for a value is obtained by subtracting the mean from the value and dividing the result by the standard deviation.
nilai standard atau z score untuk suatu nilai diperoleh dengan mengurangkan nilai tersebut dengan rata-rata dan membagi hasilnya dengan standar deviasi.
The symbol z is used for the z score.
The z score represents the number of standard deviations a data value falls above or below the mean.
z score mewakili jumlah standar deviasi nilai data jatuh di atas atau di bawah rata-rata.

For samples:
z=X−X‾sz = \dfrac{X - \overline{\rm X}}{s}
For populations:
z=X−μσz = \dfrac{X - \mu}{\sigma}

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z score - Example

A student scored 65 on a statistics exam that had a mean of 50 and a standard deviation of 10. Compute the z-score.
z=65−5010=1.5z = \dfrac{65 - 50}{10} = 1.5
That is, the score of 65 is 1.5 standard deviations above the mean.
Above - since the z-score is positive.

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Percentiles

Percentiles divide the distribution into 100 groups.
Persentil membagi distribusi menjadi 100 kelompok.
The PkP_k percentile is defined to be that numerical value such that at most k% of the values are smaller than PkP_k and at most (100 - k)% are larger than PkP_k in an ordered data set.
Persentil PkP_k didefinisikan sebagai nilai numerik sedemikian rupa sehingga paling banyak k% dari nilai-nilai lebih kecil dari PkP_k dan paling banyak (100 - k)% lebih besar dari PkP_k dalam kumpulan data yang diurutkan.
The percentile corresponding to a given value (X) is computed by using the formula:

Percentile=NumberofvaluesbelowX+0.5Totalnumberofvalues×100Percentile = \dfrac{Number of values below X + 0.5}{Total number of values} \times 100

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Percentiles - Example

A teacher gives a 20-point test to 10 students. Find the percentile rank of a score of 12.
Scores: 18, 15, 12, 6, 8, 2, 3, 5, 20, 10.
Ordered set: 2, 3, 5, 6, 8, 10, 12, 15, 18, 20.
Percentile = 6+0.510×100=65\dfrac{6 + 0.5}{10} \times 100 = 65th percentile.Student did better than 65% of the class.

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Finding the value Corresponding to a Given Percentile - Example

Find the value of the 25th percentile for the following data set: 2, 3, 5, 6, 8, 10, 12, 15, 18, 20. Procedure: Let p be the percentile and n the sample size.
1

Arrange the data in order.

the data set is already ordered.
data set: 2, 3, 5, 6, 8, 10, 12, 15, 18, 20.
2

Compute c = (np)/100.

n = 10, p = 25, so c=(10∙25)/100=2.5c = (10 \bullet 25)/100 = 2.5.
3

Check c

Hence round up to c = 3.
If c is not a whole number, round up to the next whole number.
If c is a whole number, use the value halfway between c and c+1.
4

The value of c is the position value of the required percentile.

data set: 2, 3, 5, 6, 8, 10, 12, 15, 18, 20. ⟼c=3\longmapsto c = 3
Thus, the value of the 25th percentile is the value X = 5.Find the 80th percentile.     c=10∙80100=8~~~~ c = \dfrac{10 \bullet 80}{100} = 8.Thus the value of the 80th percentile is the average of the 8th and 9th values.
Thus, the 80th percentile for the data set is (15 + 18)/2 = 16.5.

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Special Percentiles - Deciles and Quartiles

Deciles divide the data set into 10 groups.
Deciles membagi kumpulan data menjadi 10 kelompok.
Deciles are denoted by D1, D2, ..., D9D_1,~ D_2,~ ...,~ D_9 with the corresponding percentiles being P10, P20, ..., P90P_{10},~ P_{20},~ ...,~ P_{90}
Quartiles divide the data set into 4 groups.
Quartiles membagi kumpulan data menjadi 4 kelompok.
Quartiles are denoted by Q1, Q2, and Q3Q_1,~ Q_2,~ and~ Q_3 with the corresponding percentiles being P25, P50, and P75.P_{25},~ P_{50},~ and~ P_{75}.The median is the same as P50 or Q2P_{50} ~ or ~ Q_2

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Outliers and the Interquartile Range (IQR)

An outlier is an extremely high or an extremely low data value when compared with the rest of the data values.
Outlier adalah nilai data yang sangat tinggi atau sangat rendah jika dibandingkan dengan nilai data lainnya.
The Interquartile Range, IQR=Q3−Q1IQR = Q_3 - Q_1

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Outliers and the Interquartile Range (IQR) - Example

To determine whether a data value can be considered as an outlier: Given the data set 5, 6, 12, 13, 15, 18, 22, 50, can the value of 50 be considered as an outlier?
1

Compute Q1 and Q3

Q1=9, Q3=20Q_1 = 9,~ Q_3 = 20
To find Q1 and Q3, first find the median.
If the number of data values is odd, the median is the middle value.
If the number of data values is even, the median is the average of the two middle values.For the data set 5, 6, 12, 13, 15, 18, 22, 50, the median is 13, 15.
The lower quartile Q1Q_1 is the median of the lower half of the data set, which is (6 + 12)/2 = 9.
The upper quartile Q3Q_3 is the median of the upper half of the data set, which is (15 + 18)/2 = 20.
2

Find the IQR = Q3 - Q1

IQR=Q3−Q1=20−9=11IQR = Q_3 - Q_1 = 20 - 9 = 11
3

Compute (1.5)(IQR)

(1.5)(IQR)=(1.5)(11)=16.5(1.5)(IQR) = (1.5)(11) = 16.5
4

Compute Q1 - (1.5)(IQR) and Q3 + (1.5)(IQR)

Q1−(1.5)(IQR)=9−16.5=−7.5Q_1 - (1.5)(IQR) = 9 - 16.5 = -7.5
Q3+(1.5)(IQR)=20+16.5=36.5Q_3 + (1.5)(IQR) = 20 + 16.5 = 36.5
5

Compare the data value

The value of 50 is outside the range -7.5 to 36.5, and the value of 50 can be considered as an outlier.
Compare the data value (say X) with Q1 - (1.5)(IQR) and Q3 + (1.5)(IQR).If X < Q1 - (1.5)(IQR) or
if X > Q3 + (1.5)(IQR), then X is considered an outlier.