Division of Materials and Waste Management
May 2019
Calculating 95% Upper Confidence
Level (UCL)
Example Calculation of the 95% UCL for a Normal Mean
Ten samples of material are taken to demonstrate that the material meets the beneficial use
standards in Table 1 in the Guidance Document. The samples are obtained using a simple random
sampling design. For each constituent tested, you must obtain 95% UCL. Analysis of the samples
for lead generated the following results: 160, 175, 210, 220, 230, 240, 245, 270, 310, and 380
ppm. The limit for lead is 300 ppm.
Step 1: Mean and Standard Deviation Calculation
Assuming a normal model is acceptable (using the Shapiro-Wilk test), the mean and standard
deviation should be calculated. The mean and standard deviation can be obtained using statistical
software or by hand, using the following equations.
Standard deviation for a sample:
1) Subtract the Mean from each data point and square the result:
Data Point 1:
Data Point 2:
Data Point 3:
Data Point 4:
Data Point 5:
Data Point 6:
Data Point 7:
Data Point 8:
Data Point 9:
Data Point 10:
2) Divide the sum of those squared differences by the number of data points minus 1:
3) Take the square root of that number to obtain the standard deviation:
Step 2: T-Value and 95% UCL Calculation
1) Find your T-Value from Table 1 for the number of samples taken:
# of
samples
minus 1 (n-1)
T Value
for 95% UCL
# of
samples
minus 1 (n-1)
T Value
for 95% UCL
1
6.314
11
1.796
2
2.920
12
1.782
3
2.353
13
1.771
4
2.132
14
1.761
5
2.015
15
1.753
6
1.943
16
1.746
7
1.895
17
1.740
8
1.860
18
1.734
9
1.833
19
1.729
10
1.812
20
1.725
Table 1: T Values
2) The UCL is calculated as follows:
Step 3: Compare 95% UCL to the Table of Constituents
Compare the 95% UCL calculated in Step 2 to the limit in the table of constituents for that specific
constituent. By showing that the 95% UCL for the sample, which is 281 ppm, is less than the 300-
ppm limit for lead, we can conclude with at least 95% confidence that the mean concentration
of the constituent in the material is less than 300 ppm.
What Should I do with My Nondetect Results?
U.S. EPA’s ProUCL Statistical Software for
Environmental Applications for Data Sets with and
without Nondetect (ND) Observations was
specifically designed to evaluate environmental data
sets with limited sample numbers, ND results, and
skewed data. It can be run on environmental data sets with and without ND data samples.
ProUCL requires no formal background in statistics, but some statistical training is helpful to
understand the assumptions and input requirements for statistical tests used in decision making.
Input data sets are straightforward, requiring columns of detected values for contaminants and
whether each value is a detect or a ND at the quantitation limit. Desired statistical tests can be
selected from drop-down menus, and relevant options from subsequent menus. Data can be
evaluated for fit to normal, lognormal, or gamma distributions. Outputs include
recommendations, cautions, and cited references.
For data sets containing NDs with multiple detection limits (DLs) or reporting limits (RLs), ProUCL
has several estimation methods including the Kaplan-Meier (KM) method, regression on order
statistics (ROS) methods and substitution methods (e.g., replacing NDs by DL, DL/2). In addition
to computing general statistics, ProUCL has goodness-of-fit (GOF) tests for normal, lognormal,
and gamma distributions, and parametric and nonparametric methods including bootstrap
methods for skewed data sets for computation of decision-making statistics. Ohio EPA highly
recommends using ProUCL for all statistical computations. U.S. EPA offers tutorials on ProUCL.
The following information was taken from the ProUCL 5.1 user’s guide:
The user informs the program about the status of a variable consisting of NDs. No qualifiers or
flags (e.g., J, B, U, UJ, X, or <DL or RL) should be entered in data files with ND observations. You
can copy and paste your data from an EXCEL spreadsheet into a Work Sheet xls in ProUCL.
Enter the data for variables with ND values in two columns. One column should consist of
numerical values of detected observations and numerical values of detection limits (or reporting
limits) associated with observations reported as NDs; and the second column represents their
detection status consisting of only 0 (for ND values) and 1 (for detected values) values. The name
of the corresponding variable representing the detection status should start with d_, or D_ (not
Resources for Statistical Analysis
•
Calculating Upper Confidence Limits for
Exposure Point Concentrations at Hazardous
Waste Sites
•
ProUCL Version Technical Guide.
case sensitive) and the variable name. The detection status column with variable name starting
with a D_ (or a d_) should have only two values: 0 for ND values, and 1 for detected observations.
See the example below, the header name, D_Arsenic is used for the variable, Arsenic having ND
observations. The variable D_Arsenic contains a 1 if the corresponding Arsenic value represents a
detected entry and contains a 0 if the corresponding entry represents a ND entry. If this format is
not followed, the program will not recognize that the data set has NDs.
The General Statistics/With NDs option also provides simple statistics (e.g., % NDs, Max detect,
Min detect, Mean) based upon detected values. The statistics computed in log-scale (e.g., sd of
log-transformed detected values) may help a user to determine the degree of skewness (e.g., mild,
moderate, high) of a data set based upon detected values. These statistics may also help the user
to choose the most appropriate method (e.g., KM bootstrap-t UCL or KM percentile bootstrap
UCL) to compute UCLs, UPLs, and other limits used to compute decision statistics.
Figure A: ProUCL Example Data
For most beneficial use permits, the permittee is required to determine that the 95% UCL of the
mean for each constituent in the beneficial use byproduct does not exceed its constituent
concentration limit specified in the permit. DMWM highly recommends using ProUCL to
calculate the 95% UCL, especially when you have NDs in your sample results. Below is an example
of the output from selecting UCLs/EPCs and “All” from the drop-down menu.
Arsenic
8 4
1
0 0.203
0.735 0
0.325 0.115
1.597 1.291
0.664
0.818
0.359
0.283
0.421 0.448
0.429
0.392 0.421
N/A N/A
N/A N/A
N/A
0.547 0.703
0.92 1.345
0.703
Maximum
SD
Number of Distinct Observations
Number of Missing Observations
Mean
Median
Std. Error of Mean
General Statistics
Total Number of Observations
Minimum
Note: Sample size is small (e.g., <10), if data are collected using ISM approach, you should use
guidance provided in ITRC Tech Reg Guide on ISM (ITRC, 2012) to compute statistics of interest.
For example, you may want to use Chebyshev UCL to estimate EPC (ITRC, 2012).
Chebyshev UCL can be computed using the Nonparametric and All UCL Options of ProUCL 5.1
Coefficient of Variation
Skewness
Data Not Normal at 5% Significance Level
Normal GOF Test
Shapiro Wilk Test Statistic
5% Shapiro Wilk Critical Value
Lilliefors Test Statistic
5% Lilliefors Critical Value
Shapiro Wilk GOF Test
Data Not Normal at 5% Significance Level
Lilliefors GOF Test
Data Not Normal at 5% Significance Level
Gamma Statistics Not Available
Lognormal Statistics Not Available
Assuming Normal Distribution
95% Normal UCL
95% Student's-t UCL
95% UCLs (Adjusted for Skewness)
95% Adjusted-CLT UCL (Chen-1995)
95% Modified-t UCL (Johnson-1978)
Note: Suggestions regarding the selection of a 95% UCL are provided to help the user to select the most appropriate 95% UCL.
Recommendations are based upon data size, data distribution, and skewness.
These recommendations are based upon the results of the simulation studies summarized in Singh, Maichle, and Lee (2006).
However, simulations results will not cover all Real World data sets; for additional insight the user may want to consult a statistician.
95% Hall's Bootstrap UCL
95% BCA Bootstrap UCL
90% Chebyshev(Mean, Sd) UCL
97.5% Chebyshev(Mean, Sd) UCL
Suggested UCL to Use
95% Chebyshev (Mean, Sd) UCL
95% Percentile Bootstrap UCL
95% Chebyshev(Mean, Sd) UCL
99% Chebyshev(Mean, Sd) UCL
Nonparametric Distribution Free UCL Statistics
Data do not follow a Discernible Distribution (0.05)
Nonparametric Distribution Free UCLs
95% CLT UCL
95% Standard Bootstrap UCL
95% Jackknife UCL
95% Bootstrap-t UCL
