GROUPED PERCENTILE
GROUPED QUANTILE

GROUPED PERCENTILE (LET)
GROUPED QUANTILE (LET)

Let Subcommand

Compute the percentile (or quantile) for grouped data.

The PERCENTILE (QUANTILE) command is used to compute a specified percentile (quantile) of a univariate response variable. However, there may be cases where your data is only available as a frequency table. For these cases, the GROUPED PERCENTILE (QUANTILE) command can be used to compute a specified percentile (quantile).

If the groups are equi-spaced and the underlying data is continuous, then the group-id variable may be given as the mid-points of the group intervals. If the groups are not equi-spaced or if the underlying data is not continuous, then two group-id variables should be given: the first specifies the lower limits for the groups and the second specifies the upper limits for the groups.

To compute the i-th percentile for grouped data, Dataplot does the following

1. If only the group mid-points are provided, these are first converted to lower and upper class boundaries. The class width is computed as XMID(2) - XMID(1) (where XMID is the variable containing the group mid-points). Then the lower boundary for class i is XMID(i) - (class width/2) and the upper boundary for class i is XMID(i) + (class width/2).

2. The percentiles corresponding to the upper class boundaries are computed (100*F(i)/NTOTAL where F(i) is the cumulative frequency of class i and NTOTAL is the sum of the frequencies for all classes).

   LI>Determine which class the specified percentile falls into. Call this ICLASS.

3. Then use the following formula to compute the specified percentile
   \( p_{i} = l + \frac{(iN/100) - F(ICLASS-1)}{f(ICLASS)} C \)

   where

   i	=	the specified percentile (a value between 0 and 100)
   l	=	the lower limit of the ICLASS group
   N	=	the sum of the frequencies for all groups
   F	=	the cumulative frequuency for the ICLASS - 1 group
   f	=	the frequuency for the ICLASS group
   C	=	the class width for the ICLASS group

Note that this formula implicitly assumes that the frequencies are uniformly distributed within a class.

LET <par> = <percentile> GROUPED PERCENTILE <freq> <xmid>
                        <SUBSET/EXCEPT/FOR qualification>
where <percentile> is a number or parameter with a value between 0 and 100 that specifies the desired percentile;
            <freq> is the variable containing the frequencies;
            <xmid> is the variable containing the mid-points of the groups;
            <par> is a parameter where the computed percentile is saved;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This is used for the case where you have equi-spaced intervals for the groups and the underlying data is assumed to be continuous. The class width is set to XMID(2) - XMID(1) (where XMID is the group-id variable), the lower boundary is set to XMID(i) - (class width/2) and the upper boundary is set to XMID(i) + (class width/2).

LET <par> = <percentile> GROUPED PERCENTILE <freq> <xlow> <xhigh>
                        <SUBSET/EXCEPT/FOR qualification>
where <percentile> is a number or parameter with a value between 0 and 100 that specifies the desired percentile;
            <freq> is the variable containing the frequencies;
            <xlow> is the variable containing the mid-points of the groups;
            <par> is a parameter where the computed percentile is saved;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This is used for the case where you have unequally spaced intervals or the underlying data may not be continuous. The class width for the i-th group is XHIGH(i) - XLOW(i) (where XHIGH and XLOW are the upper and lower class boundaries, respectively).

LET <par> = <quantile> GROUPED QUANTILE <freq> <xmid>
                        <SUBSET/EXCEPT/FOR qualification>
where <quantile> is a number or parameter with a value between 0 and 1 that specifies the desired quantile;
            <freq> is the variable containing the frequencies;
            <xmid> is the variable containing the mid-points of the groups;
            <par> is a parameter where the computed quantile is saved;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This is used for the case where you have equi-spaced intervals for the groups and the underlying data is assumed to be continuous. The class width is set to XMID(2) - XMID(1) (where XMID is the group-id variable), the lower boundary is set to XMID(i) - (class width/2) and the upper boundary is set to XMID(i) + (class width/2).

LET <par> = <quantile> GROUPED quantile <freq> <xlow> <xhigh>
                        <SUBSET/EXCEPT/FOR qualification>
where <quantile> is a number or parameter with a value between 0 and 1 that specifies the desired quantile;
            <freq> is the variable containing the frequencies;
            <xlow> is the variable containing the mid-points of the groups;
            <par> is a parameter where the computed quantile is saved;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This is used for the case where you have unequally spaced intervals or the underlying data may not be continuous. The class width for the i-th group is XHIGH(i) - XLOW(i) (where XHIGH and XLOW are the upper and lower class boundaries, respectively).

LET XPERC = 70 GROUPED PERCENTILE FREQ XMID
LET XPERC = 70 GROUPED PERCENTILE FREQ XLOWER XUPPER


LET P = 70
LET XPERC = P GROUPED PERCENTILE Y X
LET XPERC = P GROUPED PERCENTILE Y X1 X2


LET XQUAN = 0.70 GROUPED QUANTILE FREQ XMID
LET XQUAN = 0.90 GROUPED QUANTILE FEEQ XLOWER XUPPER


LET QUANT = 0.80
LET XQUAN = P GROUPED QUANTILE Y X
LET XQUAN = P GROUPED QUANTILE Y X1 X2

Dataplot statistics can be used in a number of commands. For details, enter
HELP STATISTICS

None

The following can also be used
LET P100 =
LET XPERRC = GROUPED PERCENTILE Y X
LET XPERRC = GROUPED PERCENTILE Y X1 X2


LET P100 =
LET XQUAN = GROUPED QUANTILE Y X
LET XQUAN = GROUPED QUANTILE Y X1 X2

PERCENTILE	=	Compute a specified percentile of a response variable.
QUANTILE	=	Compute a specified quantile of a response variable.

Selby (1974), "CRC Standard Mathematical Tables", 22nd Edition, CRC Press, p. 572.

Frequency Data

2024/09

 
let y = double exponential random numbers for i = 1 1 1000
set histogram empty bins off
set histogram class width normal corrected
let y2 x2 = binned y
.
LET P05  =  5 GROUPED PERCENTILE Y2 X2
LET P10  = 10 GROUPED PERCENTILE Y2 X2
LET P20  = 20 GROUPED PERCENTILE Y2 X2
LET P50  = 50 GROUPED PERCENTILE Y2 X2
LET P80  = 80 GROUPED PERCENTILE Y2 X2
LET P90  = 90 GROUPED PERCENTILE Y2 X2
LET P95  = 95 GROUPED PERCENTILE Y2 X2
PRINT P05 P10 P20 P50 P80 P90 P95
    

The following output is generated

 
 PARAMETERS AND CONSTANTS--

    P05     --       -2.28593
    P10     --       -1.59587
    P20     --       -0.85970
    P50     --       -0.03325
    P80     --        0.77356
    P90     --        1.52499
    P95     --        2.10227