# question on five number summary & quantile.

i know that in five number summary :

25% of a data set lies between Min & 1st quartile.

50% of a data set lies between Min & 2nd quartile, that is, Median.

75% of a data set lies between Min & 3rd quartile.

& 100% of a data set lies between Min & Max.

But my cocept is contradicting as i run a data set in R Programming Language.

while i wrote the command fivenum(x) & quantile(x) the percentiles are not matching that is, "the 1st quartile of fivenum(x)" is not "the the value below which 25% of the data fall in quantile(x)"

the commands that i have run in R console:

y <- c(7,12,9,1)

fivenum(y)

[1] 1.0 4.0 8.0 10.5 12.0

quantile(y)

0% 25% 50% 75% 100%

1.00 5.50 8.00 9.75 12.00

by which points am i being misleading?

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$\frac{n+1}{2}\times 25\% =1.25\\ \frac{n+1}{2}\times 75\% =3.75$

The fivenum command takes the $25^{th}$ percentile as the number half way between $1$ and $7$ and the $75^{th}$ as the number half way between $9$ and $12$, whereas the quantile command may have been an attempt to produce the numbers $0.25$ of the way between $1$ and $7$ and $0.75$ of the way between $9$ and $12$ that contained an error and ended up giving the numbers $0.25$ of the way between 7 and 1 and $0.75$ of the way between $12$ and $9$.

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I see a few getting confused between the quartiles given by summary() and fivenum().

Firstly - summary() gives the following summary statistics:

[Minimum] [1st Quartile] [Median] [Mean] [3rd Quartile] [Maximum]


where as,

fivenum() returns Tukey's five number summary i.e.

[Minimum] [Lower hinge] [Median] [Upper hinge] [Maximum]


Here comes the confusion - what's the difference between the quartiles and lower/upper hinges?

Let me explain with an example - Try this in R:

First - with a vector "y" of odd number of values (5 in this example)

> y=c(2, 5, 8, 15, 8)
> summary(y)    Min. 1st Qu.  Median    Mean 3rd Qu.   Max.
>               2.0     5.0     8.0     7.6     8.0    15.0
> fivenum(y)
> [1]  2  5  8  8 15


As you can see results are the same, except that summary() gave the mean value in addition to what fivenum() displayed.

Now I am including just one more variable (new value: 12) and defined this vector as "z"; Note that the vector count is now even (6 values)

> z=c(2, 5, 8, 12, 15, 18)
> summary(z)
Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
2.00    5.75   10.00   10.00   14.25   18.00
> fivenum(z)
[1]  2  5 10 15 18


Now see the difference - while summary() gave you the quartiles (calculated values based on quartile or percentile formulae), this is what fivenum() does - making it really simple - you can do it without math calculations:

Description of fivenum() output:

Firstly, min, median & max values given by fivenum() are straight forward!

Lower Hinge = median of the values to the left of (MEDIAN OF ALL VALUES) = median of the values that are less than 10 = median of (2,5,8) = 5

Upper hinge = median of the values to the right of (MEDIAN OF ALL VALUES) = median of the values that are greater than 10 = median of (12,15,18) = 15

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