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, y, as follows:

Example:

N = 3

y1 = 10.60

y2 = 11.91

yN = y3 = 11.09

y = 11.20

Example:

N = 3

y1 = 10.60

y2 = 11.91

yN = y3 = 11.09

y = 11.20

σy = 0.6619

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

Example:

N = 3

y1 = 10.60

y2 = 11.91

yN = y3 = 11.09

rmsy = 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 ref [=E T=0362]i . If you use a standard quantity with a single value, y ref [=E T=0362]i would be constant. Calculate an accuracy value as follows:

Example:

yref = 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, ν, as follows:

Example:

yref = 1205.3

y = 1123.8

σref = 9.399

σy = 10.583

Nref = 11

N= 7

t = 16.63

σref = 9.399

σy = 10.583

Nref = 11

N = 7

ν = 11.76

(2) For a paired t -test, calculate the t statistic and its number of degrees of freedom, ν, as follows, noting that the εi are the errors (e.g., differences) between each pair of y refi and y i :

Example:

ε = −0.12580

N = 16

σε = 0.04837

t = 10.403

ν = N − 1

Example:

N = 16

ν = 16 − 1

ν = 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 package contains a TINV function that returns results equivalent to § 1065.602 Table 1 and may be used in place of Table 1.

Table 1 of § 1065.602 —Critical t Values Versus Number of Degrees of Freedom, ν 1

ν	Confidence
ν	90%	95%
1	6.314	12.706
2	2.920	4.303
3	2.353	3.182
4	2.132	2.776
5	2.015	2.571
6	1.943	2.447
7	1.895	2.365
8	1.860	2.306
9	1.833	2.262
10	1.812	2.228
11	1.796	2.201
12	1.782	2.179
13	1.771	2.160
14	1.761	2.145
15	1.753	2.131
16	1.746	2.120
18	1.734	2.101
20	1.725	2.086
22	1.717	2.074
24	1.711	2.064
26	1.706	2.056
28	1.701	2.048
30	1.697	2.042
35	1.690	2.030
40	1.684	2.021
50	1.676	2.009
70	1.667	1.994
100	1.660	1.984
1000	1.645	1.960
1 Use linear interpolation to establish values not shown here.

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

Example:

F = 1.268

(1) For a 90% confidence F -test, use Table 2 of this section 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 Table 3 of this section to compare F to the F crit95 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.

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[Please see PDF for image: ER13JY05.017 ]

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[Please see PDF for image: ER13JY05.018 ]

(h) Slope. Calculate a least-squares regression slope, a 1y, as follows:

Example:

N = 6000

y1 = 2045.8

y = 1051.1

yref 1 = 2045.0

yref = 1055.3

a1y = 1.0110

(i) Intercept. Calculate a least-squares regression intercept, a 0y, as follows:

Example:

y = 1050.1

a1y = 1.0110

yref = 1055.3

a0y = 1050.1 − (1.0110 · 1055.3)

a0y = −16.8083

(j) Standard estimate of error. Calculate a standard estimate of error, SEE, as follows:

Example:

N = 6000

y1 = 2045.8

a0y = −16.8083

a1y = 1.0110

yrefi= 2045.0

SEEy = 5.348

(k) Coefficient of determination. Calculate a coefficient of determination, r 2, as follows:

Example:

N = 6000

y1 = 2045.8

a0y = −16.8083

a1y = 1.0110

yrefi = 2045.0

y = 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 NOX concentration from a turbocharged heavy-duty compression-ignition engine at a NOX 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, P ref.

(ii) Based on your engine design, estimate maximum power, P max, the design speed at maximum power, f 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, P frict. 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 (2-stroke or 4-stroke), N stroke to estimate the maximum raw exhaust molar flow rate, n exhmax.

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

Example:

eNOx = 2.5 g/(kW·hr)

Wref = 11.883 kW·hr

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

Δtdutycycle = 20 min = 1200 s

Pref = 35.65 kW

Pfrict = 15%

Pmax = 125 kW

pmax = 300 kPa = 300000 Pa

Vdisp = 3.0 l = 0.0030 m3

fnmax = 2800 rev/min = 46.67 rev/s

Nstroke = 4 1/rev

ηV = 0.9

R = 8.314472 J/(mol·K)

Tmax = 348.15 K

nexhmax = 6.53 mol/s

xexp = 189.4 µmol/mol

(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:

(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:

eNMHC = 1.5 g/(kW·hr)

Wref = 5.389 kW·hr

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

ndexh = 6.021 mol/s

Δtdutycycle = 30 min = 1800 s

xNMHC = 53.8 µmol/mol

Code of Federal Regulations

[70 FR 40516, July 13, 2005, as amended at 73 FR 37324, June 30, 2008; 75 FR 23044, Apr. 30, 2010]

CFR

1065.602—Statistics.

Code of Federal Regulations

Code of Federal Regulations

Code of Federal Regulations