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Dataplot Vol 1 Vol 2

SD CONFIDENCE LIMITS

Name:
    SD CONFIDENCE LIMITS
Type:
    Analysis Command
Purpose:
    Generates a confidence interval for the standard deviation.
Description:
    Given a sample of n observations with standard deviation s, the two-sided confidence interval for the standard deviation is

      \( \mbox{lower confidence limit} = s \sqrt{\frac{n-1}{\chi^{2}_{(1-\alpha/2;n-1)}}} \)

      \( \mbox{upper confidence limit} = s \sqrt{\frac{n-1}{\chi^{2}_{(\alpha/2;n-1)}}} \)

    with \( \chi^{2} \) denoting the percent point function of the chi-square distribution. In these formulas, \( \alpha \) is less than 0.5 (i.e., for a 95% confidence interval, we are using \( \alpha \) = 0.05).

    The one-sided lower confidence limit is

      \( \mbox{lower confidence limit} = s \sqrt{\frac{n-1}{\chi^{2}_{(1-\alpha;n-1)}}} \)

    The one-sided upper confidence limit is

      \( \mbox{upper confidence limit} = s \sqrt{\frac{n-1}{\chi^{2}_{(\alpha;n-1)}}} \)

    This confidence interval is based on the assumption that the underlying data is approximately normally distributed. The confidence interval for the standard deviation is highly sensitive to non-normality in the data. It is recommended that the original data be tested for normality before using these normal based intervals. If the data is not approximately normal, an alternative is to use the command

      BOOTSTRAP STANDARD DEVIATION PLOT Y
Syntax 1:
    <LOWER/UPPER> SD CONFIDENCE LIMITS <y>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y> is the response variable;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    If LOWER is specified, a one-sided lower confidence limit is returned. If UPPER is specified, a one-sided upper confidence limit is returned. If neither is specified, a two-sided limit is returned.

    This syntax supports matrix arguments for the response variable.

Syntax 2:
    MULTIPLE <LOWER/UPPER> SD CONFIDENCE LIMITS <y1> ... <yk>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y1> .... <yk> is a list of 1 to 30 response variables;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax will generate a confidence interval for each of the response variables. The word MULTIPLOT is optional. That is,

      MULTIPLE SD CONFIDENCE LIMITS Y1 Y2 Y3

    is equivalent to

      SD CONFIDENCE LIMITS Y1 Y2 Y3

    If LOWER is specified, a one-sided lower confidence limit is returned. If UPPER is specified, a one-sided upper confidence limit is returned. If neither is specified, a two-sided limit is returned.

    This syntax supports matrix arguments for the response variables.

Syntax 3:
    REPLICATED <LOWER/UPPER> SD CONFIDENCE LIMITS <y> <x1> ... <xk>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y> is the response variable;
                <x1> .... <xk> is a list of 1 to 6 group-id variables;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax performs a cross-tabulation of the <x1> ... <xk> and generates a confidence interval for each unique combination of the cross-tabulated values. For example, if X1 has 3 levels and X2 has 2 levels, six confidence intervals will be generated.

    If LOWER is specified, a one-sided lower confidence limit is returned. If UPPER is specified, a one-sided upper confidence limit is returned. If neither is specified, a two-sided limit is returned.

    This syntax does not support matrix arguments.

Examples:
    SD CONFIDENCE LIMITS Y1
    SD CONFIDENCE LIMITS Y1 SUBSET TAG > 2
    SD CONFIDENCE LIMITS Y1 TO Y5
    REPLICATED SD CONFIDENCE LIMITS Y X
Note:
    A table of confidence limits is printed for alpha levels of 50.0, 80.0, 90.0, 95.0, 99.0, and 99.9.
Note:
    In addition to the STANDARD DEVIATION CONFIDENCE LIMIT command, the following commands can also be used:

      LET ALPHA = 0.05

      LET A = LOWER STANDARD DEVIATION CONFIDENCE LIMIT Y
      LET A = UPPPER STANDARD DEVIATION CONFIDENCE LIMIT Y
      LET A = ONE SIDED LOWER STANDARD DEVIATION CONFIDENCE
                      LIMIT Y
      LET A = ONE SIDED UPPER STANDARD DEVIATION CONFIDENCE
                      LIMIT Y

      LET A = SUMMARY LOWER STANDARD DEVIATION CONFIDENCE
                      LIMIT YSD N
      LET A = SUMMARY UPPPER STANDARD DEVIATION CONFIDENCE
                      LIMIT YSD N
      LET A = SUMMARY ONE SIDED LOWER STANDARD DEVIATION
                      CONFIDENCE LIMIT YSD N
      LET A = SUMMARY ONE SIDED UPPER STANDARD DEVIATION
                      CONFIDENCE LIMIT YSD N

    The first command specifies the significance level. The next four commands are used when you have raw data. The last four commands are used when only summary data ( standard deviation, sample size) is available.

    In addition to the above LET command, built-in statistics are supported for about 20 different commands (enter HELP STATISTICS for details).

Default:
    None
Synonyms:
    STANDARD DEVIATION CONFIDENCE INTERVAL is a synonym for STANDARD DEVIATION CONFIDENCE LIMITS

    SD CONFIDENCE LIMIT is a synonym for STANDARD DEVIATION CONFIDENCE LIMIT

Related Commands: Reference:
    Hahn and Meeker (1991), "Statistical Intervals: A Guide for Practitioners," Wiley, pp. 55-56.
Applications:
    Confirmatory Data Analysis
Implementation Date:
    2013/04
Program 1:
     
    SKIP 25
    READ ZARR13.DAT Y
    SET WRITE DECIMALS 5
    .
    SD CONFIDENCE LIMITS Y
    LOWER SD CONFIDENCE LIMITS Y
    UPPER SD CONFIDENCE LIMITS Y
        
    The following output is generated
                Two-Sided Confidence Limits for the SD
     
    Response Variable: Y
     
    Summary Statistics:
    Number of Observations:                             195
    Sample Mean:                                    9.26146
    Sample Standard Deviation:                      0.02278
     
     
     
    Two-Sided Confidence Limits for the SD
    ------------------------------------------
      Confidence          Lower          Upper
       Value (%)          Limit          Limit
    ------------------------------------------
            50.0        0.02206        0.02363
            80.0        0.02141        0.02440
            90.0        0.02104        0.02487
            95.0        0.02072        0.02530
            99.0        0.02013        0.02617
            99.9        0.01948        0.02725
     
     
                One-Sided Lower Confidence Limits for the SD
     
    Response Variable: Y
     
    Summary Statistics:
    Number of Observations:                             195
    Sample Mean:                                    9.26146
    Sample Standard Deviation:                      0.02278
     
     
     
    One-Sided Lower Confidence Limits for the SD
    ---------------------------
      Confidence          Lower
       Value (%)          Limit
    ---------------------------
            50.0        0.02282
            80.0        0.02188
            90.0        0.02141
            95.0        0.02104
            99.0        0.02037
            99.9        0.01966
     
     
                One-Sided Upper Confidence Limits for the SD
     
    Response Variable: Y
     
    Summary Statistics:
    Number of Observations:                             195
    Sample Mean:                                    9.26146
    Sample Standard Deviation:                      0.02278
     
     
     
    One-Sided Upper Confidence Limits for the SD
    ---------------------------
      Confidence          Upper
       Value (%)          Limit
    ---------------------------
            50.0        0.02282
            80.0        0.02384
            90.0        0.02440
            95.0        0.02487
            99.0        0.02581
            99.9        0.02694
        
Program 2:
     
    SKIP 25
    READ GEAR.DAT Y X
    .
    SET WRITE DECIMALS 5
    REPLICATED SD CONFIDENCE LIMITS Y X
        
    The following output is generated
                Two-Sided Confidence Limits for the SD
     
    Response Variable: Y
    Factor Variable 1: X                     1.00000
     
    Summary Statistics:
    Number of Observations:                  10
    Sample Mean:                             0.99800
    Sample Standard Deviation:               0.00435
     
     
     
    Two-Sided Confidence Limits for the SD
    ------------------------------------------
      Confidence          Lower          Upper
       Value (%)          Limit          Limit
    ------------------------------------------
            50.0        0.00386        0.00537
            80.0        0.00340        0.00639
            90.0        0.00317        0.00715
            95.0        0.00299        0.00793
            99.0        0.00268        0.00990
            99.9        0.00239        0.01323
     
     
                Two-Sided Confidence Limits for the SD
     
    Response Variable: Y
    Factor Variable 1: X                     2.00000
     
    Summary Statistics:
    Number of Observations:                  10
    Sample Mean:                             0.99910
    Sample Standard Deviation:               0.00522
     
     
     
    Two-Sided Confidence Limits for the SD
    ------------------------------------------
      Confidence          Lower          Upper
       Value (%)          Limit          Limit
    ------------------------------------------
            50.0        0.00464        0.00644
            80.0        0.00408        0.00767
            90.0        0.00380        0.00858
            95.0        0.00359        0.00952
            99.0        0.00322        0.01188
            99.9        0.00287        0.01588
     
     
                Two-Sided Confidence Limits for the SD
     
    Response Variable: Y
    Factor Variable 1: X                     3.00000
     
    Summary Statistics:
    Number of Observations:                  10
    Sample Mean:                             0.99540
    Sample Standard Deviation:               0.00398
     
     
     
    Two-Sided Confidence Limits for the SD
    ------------------------------------------
      Confidence          Lower          Upper
       Value (%)          Limit          Limit
    ------------------------------------------
            50.0        0.00354        0.00491
            80.0        0.00311        0.00584
            90.0        0.00290        0.00654
            95.0        0.00274        0.00726
            99.0        0.00246        0.00906
            99.9        0.00219        0.01211
     
     
                Two-Sided Confidence Limits for the SD
     
    Response Variable: Y
    Factor Variable 1: X                     4.00000
     
    Summary Statistics:
    Number of Observations:                  10
    Sample Mean:                             0.99820
    Sample Standard Deviation:               0.00385
     
     
     
    Two-Sided Confidence Limits for the SD
    ------------------------------------------
      Confidence          Lower          Upper
       Value (%)          Limit          Limit
    ------------------------------------------
            50.0        0.00343        0.00476
            80.0        0.00302        0.00566
            90.0        0.00281        0.00634
            95.0        0.00265        0.00703
            99.0        0.00238        0.00878
            99.9        0.00212        0.01173
     
     
                Two-Sided Confidence Limits for the SD
     
    Response Variable: Y
    Factor Variable 1: X                     5.00000
     
    Summary Statistics:
    Number of Observations:                  10
    Sample Mean:                             0.99190
    Sample Standard Deviation:               0.00758
     
     
     
    Two-Sided Confidence Limits for the SD
    ------------------------------------------
      Confidence          Lower          Upper
       Value (%)          Limit          Limit
    ------------------------------------------
            50.0        0.00674        0.00936
            80.0        0.00593        0.01114
            90.0        0.00553        0.01247
            95.0        0.00521        0.01384
            99.0        0.00468        0.01726
            99.9        0.00417        0.02306
     
     
                Two-Sided Confidence Limits for the SD
     
    Response Variable: Y
    Factor Variable 1: X                     6.00000
     
    Summary Statistics:
    Number of Observations:                  10
    Sample Mean:                             0.99880
    Sample Standard Deviation:               0.00989
     
     
     
    Two-Sided Confidence Limits for the SD
    ------------------------------------------
      Confidence          Lower          Upper
       Value (%)          Limit          Limit
    ------------------------------------------
            50.0        0.00879        0.01221
            80.0        0.00774        0.01453
            90.0        0.00721        0.01626
            95.0        0.00680        0.01805
            99.0        0.00611        0.02252
            99.9        0.00545        0.03009
     
     
                Two-Sided Confidence Limits for the SD
     
    Response Variable: Y
    Factor Variable 1: X                     7.00000
     
    Summary Statistics:
    Number of Observations:                  10
    Sample Mean:                             1.00150
    Sample Standard Deviation:               0.00788
     
     
     
    Two-Sided Confidence Limits for the SD
    ------------------------------------------
      Confidence          Lower          Upper
       Value (%)          Limit          Limit
    ------------------------------------------
            50.0        0.00700        0.00973
            80.0        0.00617        0.01158
            90.0        0.00575        0.01296
            95.0        0.00542        0.01438
            99.0        0.00487        0.01794
            99.9        0.00434        0.02397
     
     
                Two-Sided Confidence Limits for the SD
     
    Response Variable: Y
    Factor Variable 1: X                     8.00000
     
    Summary Statistics:
    Number of Observations:                  10
    Sample Mean:                             1.00040
    Sample Standard Deviation:               0.00363
     
     
     
    Two-Sided Confidence Limits for the SD
    ------------------------------------------
      Confidence          Lower          Upper
       Value (%)          Limit          Limit
    ------------------------------------------
            50.0        0.00322        0.00448
            80.0        0.00284        0.00533
            90.0        0.00265        0.00597
            95.0        0.00249        0.00662
            99.0        0.00224        0.00826
            99.9        0.00200        0.01104
     
     
                Two-Sided Confidence Limits for the SD
     
    Response Variable: Y
    Factor Variable 1: X                     9.00000
     
    Summary Statistics:
    Number of Observations:                  10
    Sample Mean:                             0.99830
    Sample Standard Deviation:               0.00414
     
     
     
    Two-Sided Confidence Limits for the SD
    ------------------------------------------
      Confidence          Lower          Upper
       Value (%)          Limit          Limit
    ------------------------------------------
            50.0        0.00368        0.00511
            80.0        0.00324        0.00608
            90.0        0.00302        0.00681
            95.0        0.00285        0.00755
            99.0        0.00256        0.00942
            99.9        0.00228        0.01259
     
     
                Two-Sided Confidence Limits for the SD
     
    Response Variable: Y
    Factor Variable 1: X                     10.00000
     
    Summary Statistics:
    Number of Observations:                  10
    Sample Mean:                             0.99480
    Sample Standard Deviation:               0.00533
     
     
     
    Two-Sided Confidence Limits for the SD
    ------------------------------------------
      Confidence          Lower          Upper
       Value (%)          Limit          Limit
    ------------------------------------------
            50.0        0.00474        0.00658
            80.0        0.00417        0.00783
            90.0        0.00389        0.00877
            95.0        0.00367        0.00973
            99.0        0.00329        0.01214
            99.9        0.00294        0.01622
        
Program 3:
     
    .  Following example from Hahn and Meeker's book.
    .
    let ymean = 50.10
    let ysd   = 1.31
    let n1    = 5
    let alpha = 0.05
    .
    set write decimals 5
    let slow1 = summary lower sd confidence limits ysd n1
    let supp1 = summary upper sd confidence limits ysd n1
    let slow2 = summary one sided lower sd confidence limits ysd n1
    let supp2 = summary one sided upper sd confidence limits ysd n1
    print slow1 supp1 slow2 supp2
        
    The following output is generated.
     PARAMETERS AND CONSTANTS--
    
        SLOW1   --        0.78486
        SUPP1   --        3.76436
        SLOW2   --        0.85059
        SUPP2   --        3.10779
        

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Date created: 04/15/2013
Last updated: 11/05/2015

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