Statistics.

1. Law
2. CFR 40
3. Protection of Environment
4. Environmental Protection Agency
5. Air Pollution Controls
6. Engine-Testing Procedures
7. Calculations and Data Requirements
8. Statistics.

§ 1065.602 Statistics.

(a) Overview. This section contains equations and example calculations for statistics that are specified in this part. In this section we use the letter “y” to denote a generic measured quantity, the superscript over-bar “^-” to denote an arithmetic mean, and the subscript “_ref” to denote the reference quantity being measured.

(b) Arithmetic mean. Calculate an arithmetic mean, ȳ, as follows:

Example:

N = 3

y₁ = 10.60

y₂ = 11.91

y_N = y₃ = 11.09

ȳ = 11.20

(c) Standard deviation. Calculate the standard deviation for a non-biased (e.g., N-1) sample, σ, as follows:

Example:

N = 3

y₁ = 10.60

y₂ = 11.91

y_N = y₃ = 11.09

ȳ = 11.20

σ_y = 0.6619

(d) Root mean square. Calculate a root mean square, rms_y, as follows:

Example:

N = 3

y₁ = 10.60

y₂ = 11.91

y_N = y₃ = 11.09

rms_y = 11.21

(e) Accuracy. Determine accuracy as described in this paragraph (e). Make multiple measurements of a standard quantity to create a set of observed values, y_i, and compare each observed value to the known value of the standard quantity. The standard quantity may have a single known value, such as a gas standard, or a set of known values of negligible range, such as a known applied pressure produced by a calibration device during repeated applications. The known value of the standard quantity is represented by y_refi. If you use a standard quantity with a single value, y_refi would be constant. Calculate an accuracy value as follows:

Example:

y_ref = 1800.0

N = 3

y₁ = 1806.4

y₂ = 1803.1

y₃ = 1798.9

accuracy = 2.8

(f) t-test. Determine if your data passes a t-test by using the following equations and tables: (1) For an unpaired t-test, calculate the t statistic and its number of degrees of freedom, v, as follows:

Example:

Ȳ_ref = 1205.3

Ȳ = 1123.8

σ_ref = 9.399

σ_y = 10.583

N_ref = 11

N = 7

t = 16.63

σ_ref = 9.399

σ_y = 10.583

N_ref = 11

N = 7

v = 11.76

(2) For a paired t-test, calculate the t statistic and its number of degrees of freedom, v, as follows, noting that the ε_i are the errors (e.g., differences) between each pair of y_refi and y_i:

Example 1:

ε̄ = −0.12580

N = 16

σ_ε = 0.04837

t = 10.403

v = N−1

Example 2:

N = 16

v = 16−1

v = 15

(3) Use Table 1 of this section to compare t to the t_crit values tabulated versus the number of degrees of freedom. If t is less than t_crit, then t passes the t-test. The Microsoft Excel software has a TINV function that returns results equivalent results and may be used in place of Table 1, which follows:

(g) F-test . Calculate the F statistic as follows:

Example:

F = 1.268

(1) For a 90% confidence F-test, use the following table to compare F to the F_crit90 values tabulated versus (N−1) and (N_ref−1). If F is less than F_crit90, then F passes the F-test at 90% confidence.

(2) For a 95% confidence F-test, use the following table to compare F to the F_crit90 values tabulated versus (N−1) and (N_ref−1). If F is less than F_crit95, then F passes the F-test at 95% confidence.

(h) Slope. Calculate a least-squares regression slope, a_1y, using one of the following two methods:

(1) If the intercept floats, i.e., is not forced through zero:

Example:

N = 6000

y₁ = 2045.8

ȳ = 1050.1

y_ref1 = 2045.0

ȳ_ref = 1055.3

a_1y = 1.0110

(2) If the intercept is forced through zero, such as for verifying proportional sampling:

Example:

N = 6000

y₁ = 2045.8

y_ref1 = 2045.0

a_1y = 1.0110

(i) Intercept . For a floating intercept, calculate a least-squares regression intercept, a_0y, as follows:

Example:

ȳ = 1050.1

a_1y = 1.0110

ȳ_ref = 1055.3

a_0y = 1050.1 − (1.0110 · 1055.3)

a_0y = −16.8083

(j) Standard error of the estimate . Calculate a standard error of the estimate, SEE, using one of the following two methods:

(1) For a floating intercept:

Example:

N = 6000

y₁ = 2045.8

a_0y = −16.8083

a_1y = 1.0110

y_ref1 = 2045.0

SEE_y = 5.348

(2) If the intercept is forced through zero, such as for verifying proportional sampling:

Example:

N = 6000

y₁ = 2045.8

a_1y = 1.0110

y_ref1 = 2045.0

SEE_y = 5.347

(k) Coefficient of determination. Calculate a coefficient of determination, r_y², as follows:

Example:

N = 6000

y₁ = 2045.8

a_0y = −16.8083

a_1y = 1.0110

y_ref1 = 2045.0

ȳ = 1480.5

(l) Flow-weighted mean concentration. In some sections of this part, you may need to calculate a flow-weighted mean concentration to determine the applicability of certain provisions. A flow-weighted mean is the mean of a quantity after it is weighted proportional to a corresponding flow rate. For example, if a gas concentration is measured continuously from the raw exhaust of an engine, its flow-weighted mean concentration is the sum of the products of each recorded concentration times its respective exhaust molar flow rate, divided by the sum of the recorded flow rate values. As another example, the bag concentration from a CVS system is the same as the flow-weighted mean concentration because the CVS system itself flow-weights the bag concentration. You might already expect a certain flow-weighted mean concentration of an emission at its standard based on previous testing with similar engines or testing with similar equipment and instruments. If you need to estimate your expected flow-weighted mean concentration of an emission at its standard, we recommend using the following examples as a guide for how to estimate the flow-weighted mean concentration expected at the standard. Note that these examples are not exact and that they contain assumptions that are not always valid. Use good engineering judgment to determine if you can use similar assumptions.

(1) To estimate the flow-weighted mean raw exhaust NO_X concentration from a turbocharged heavy-duty compression-ignition engine at a NO_X standard of 2.5 g/(kW·hr), you may do the following:

(i) Based on your engine design, approximate a map of maximum torque versus speed and use it with the applicable normalized duty cycle in the standard-setting part to generate a reference duty cycle as described in § 1065.610. Calculate the total reference work, W_ref, as described in § 1065.650. Divide the reference work by the duty cycle's time interval, Δt_dutycycle, to determine mean reference power, p̄_ref.

(ii) Based on your engine design, estimate maximum power, P_max, the design speed at maximum power, ƒ_nmax, the design maximum intake manifold boost pressure, P_inmax, and temperature, T_inmax. Also, estimate a mean fraction of power that is lost due to friction and pumping, P̄. Use this information along with the engine displacement volume, V_disp, an approximate volumetric efficiency, η_V, and the number of engine strokes per power stroke (two-stroke or four-stroke), N_stroke, to estimate the maximum raw exhaust molar flow rate, ṅ_exhmax.

(iii) Use your estimated values as described in the following example calculation:

Example:

e_NOX = 2.5 g/(kW·hr)

W_ref = 11.883 kW·hr

M_NOX = 46.0055 g/mol = 46.0055·10−⁶ g/µmol

Δt_dutycycle = 20 min = 1200 s

P̄_ref = 35.65 kW

P̄_frict = 15%

P_max = 125 kW

p_max = 300 kPa = 300000 Pa

V_disp = 3.0 l = 0.0030 m³/r

f_nmax = 2800 r/min = 46.67 r/s

N_stroke = 4

η_V = 0.9

R = 8.314472 J/(mol·K)

T_max = 348.15 K

(2) To estimate the flow-weighted mean NMHC concentration in a CVS from a naturally aspirated nonroad spark-ignition engine at an NMHC standard of 0.5 g/(kW·hr), you may do the following:

(i) Based on your engine design, approximate a map of maximum torque versus speed and use it with the applicable normalized duty cycle in the standard-setting part to generate a reference duty cycle as described in § 1065.610. Calculate the total reference work, W_ref, as described in § 1065.650.

(ii) Multiply your CVS total molar flow rate by the time interval of the duty cycle, Δt_dutycycle. The result is the total diluted exhaust flow of the n_dexh.

(iii) Use your estimated values as described in the following example calculation:

Example:

e_NMHC = 1.5 g/(kW·hr)

W_ref = 5.389 kW·hr

M_NMHC = 13.875389 g/mol = 13.875389·10−⁶ g/µmol

ṅ_dexh = 6.021 mol/s

Δt_dutycycle = 30 min = 1800 s

X̄_NMHC = 53.8 µmol/mol

[86 FR 34548, June 29, 2021]

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