# Title 40

## SECTION 1065.602

### 1065.602 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,
*y* , as follows:

*N*= 3

*y*1 = 10.60

*y*2 = 11.91

*y*N =

*y*3 = 11.09

*y*= 11.20

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

*N*= 3

*y*1 = 10.60

*y*2 = 11.91

*y*N =

*y*3 = 11.09

*y*= 11.20 sy = 0.6619

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

*N*= 3

*y*1 = 10.60

*y*2 = 11.91

*y*N =

*y*3 = 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:

*y*ref = 1800.0

*N*= 3

*y*1 = 1806.4

*y*2 = 1803.1

*y*3 = 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:

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

Table 1 of § 1065.602 - Critical t Values
Versus Number of Degrees of Freedom, v ^{1}

n | Confidence | |
---|---|---|

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:

*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.

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

*N*= 6000

*y*1 = 2045.8

*y*= 1050.1

*y*ref 1 = 2045.0

*y*ref = 1055.3

*a*1y = 1.0110

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

*y*= 1050.1

*a*1y = 1.0110

*y*ref = 1055.3

*a*0y = 1050.1 − (1.0110 · 1055.3)

*a*0y = −16.8083

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

*N*= 6000

*y*1 = 2045.8

*a*0y = -16.8083

*a*1y = 1.0110

*y*ref1 = 2045.0

*SEE*y = 5.348

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

*N*= 6000

*y*1 = 2045.8

*a*0y = −16.8083

*a*1y = 1.0110

*y*refi = 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, Δ*t*dutycycle, to
determine mean reference power, *P*ref.

(ii) Based on your engine design, estimate maximum power, Pmax,
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, hV, 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, *n* 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 Δ

*t*dutycycle = 20 min = 1200 s

*P*ref = 35.65 kW

*P*frict = 15%

*P*max = 125 kW

*p*max = 300 kPa = 300,000 Pa

*V*disp = 3.0 l = 0.0030 m 3/r

*f*nmax = 2,800 r/min = 46.67 r/s

*N*stroke = 4 ηV = 0.9

*R*= 8.314472 J/(mol · K)

*T*max = 348.15 K

*n*exhmax = 6.53 mol/s

*x*exp = 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:

(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, Δ*t*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

*n*dexh = 6.021 mol/s Δ

*t*dutycycle = 30 min = 1800 s

*x*NMHC = 53.8 µmol/mol [70 FR 40516, July 13, 2005, as amended at 73 FR 37324, June 30, 2008; 75 FR 23044, Apr. 30, 2010; 76 FR 57452, Sept. 15, 2011; 79 FR 23779, Apr. 28, 2014; 81 FR 74170, Oct. 25, 2016]