Given \(n\) observations \(x_1, x_2, \ldots, x_n\) (e.g., muzzle velocities from a chronograph), the sample mean is \[\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i.\] The sample mean is an unbiased estimator of the population mean \(\mu\). Its precision improves with \(n\): the standard error of \(\bar{x}\) is \(\sigma/\sqrt{n}\), where \(\sigma\) is the population standard deviation.

The sample variance is \[s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2,\] and the sample standard deviation is \(s = \sqrt{s^2}\). The divisor \(n-1\) (Bessel’s correction) yields an unbiased estimate of the population variance \(\sigma^2\). In practical terms, \(s\) quantifies the typical deviation of a single round from the mean—a direct measure of load consistency.

For computation, the equivalent form \[s^2 = \frac{1}{n-1}\left(\sum_{i=1}^{n} x_i^2 - n\bar{x}^2\right)\] is sometimes more convenient when working from a running total on a calculator.

Extreme Spread versus Standard Deviation

Most chronographs report the extreme spread (ES), defined as \(\mathrm{ES} = x_{\max} - x_{\min}\). While intuitive, ES is a poor metric for comparing loads because it is highly sensitive to sample size: a 10-round string will almost always show a larger ES than a 5-round string from the same ammunition, even if the underlying consistency is identical. The standard deviation \(s\) does not suffer from this bias and should be preferred whenever rigorous comparison is needed.

The statistical methods in this appendix rest on the assumption that the measured quantity follows a normal (Gaussian) distribution. The assumption is sound for muzzle velocity and for case weight, where the variation arises from the superposition of many small, independent factors (powder charge tolerance, primer brisance, neck tension, ambient temperature, etc.). By the central limit theorem, such sums converge to a normal distribution regardless of the distribution of the individual factors.

Group size is the exception, and an important one. A group size is not a sum of small effects: it is the largest distance among the shots fired—an extreme-value statistic of a two-dimensional distribution. The central limit theorem does not apply to it. Its distribution is bounded below by zero and skewed to the right, and simulation of a perfectly circular dispersion gives a skewness of about \(+0.4\) at three shots and \(+0.4\) again at twenty: unlike an average, the skew does not fade as the sample grows. The tools of this appendix—confidence intervals on the mean, tolerance bounds—are built for velocity and for case weight; they must not be carried over to group sizes without further care. Chapter [ch:diagnostics] gives the practical consequence at the target.

Definition

A continuous random variable \(X\) follows the normal distribution with mean \(\mu\) and variance \(\sigma^2\), written \(X \sim \mathcal{N}(\mu,\,\sigma^2)\), if its probability density function is \[f(x) = \frac{1}{\sigma\sqrt{2\pi}}\,\exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right), \quad -\infty < x < \infty.\] The distribution is symmetric about \(\mu\) and is completely determined by its two parameters \(\mu\) and \(\sigma\).

The Standard Normal Distribution

The standard normal distribution is the special case \(Z \sim \mathcal{N}(0,\,1)\). Any normal variable can be standardized: \[Z = \frac{X - \mu}{\sigma}.\] The cumulative distribution function \(\Phi(z) = P(Z \leq z)\) cannot be expressed in closed form and is tabulated below.

Properties

The normal distribution satisfies the well-known probability rules:

• \(P(\mu - \sigma < X < \mu + \sigma) \approx 68.27\%\),

• \(P(\mu - 2\sigma < X < \mu + 2\sigma) \approx 95.45\%\),

• \(P(\mu - 3\sigma < X < \mu + 3\sigma) \approx 99.73\%\).

For the handloader, the “\(3\sigma\) rule” means that a round deviating from the mean by more than three standard deviations is expected roughly once every 370 rounds.

Standard Normal Quantile Table

The table below gives \(z_p\) such that \(P(Z \leq z_p) = p\) for the standard normal distribution. By symmetry, \(z_{1-p} = -z_p\).

Origin and Motivation

When \(\sigma\) is known, the standardized sample mean \((\bar{x} - \mu)/(\sigma/\sqrt{n})\) follows a standard normal distribution. In practice, \(\sigma\) must be replaced by the sample estimate \(s\), and the resulting quantity \[T = \frac{\bar{x} - \mu}{s/\sqrt{n}}\] no longer follows a normal distribution. William Sealy Gosset, publishing under the pseudonym “Student” in 1908, showed that \(T\) follows a distribution with heavier tails than the normal, now called Student’s \(t\)-distribution with \(\nu = n-1\) degrees of freedom.

Definition

The probability density function of the \(t\)-distribution with \(\nu\) degrees of freedom is \[f(t) = \frac{\Gamma\!\left(\frac{\nu+1}{2}\right)}{\sqrt{\nu\pi}\;\Gamma\!\left(\frac{\nu}{2}\right)}\left(1 + \frac{t^2}{\nu}\right)^{-(\nu+1)/2}, \quad -\infty < t < \infty,\] where \(\Gamma(\cdot)\) is the gamma function. Like the normal, the \(t\)-distribution is symmetric about zero, but its tails are heavier: extreme values are more probable, reflecting the additional uncertainty from estimating \(\sigma\).

Relationship to the Normal Distribution

As \(\nu \to \infty\), the \(t\)-distribution converges to the standard normal. The convergence is rapid: for \(\nu \geq 30\), the difference is small; for \(\nu \geq 100\), it is negligible for most practical purposes. Table [tab:t_quantiles] illustrates this convergence.

Quantile Table

Table [tab:t_quantiles] gives the upper quantiles \(t_{p,\,\nu}\) such that \(P(T \leq t_{p,\,\nu}) = p\) for selected values of \(p\) and \(\nu\). The column \(p = 0.975\) is used for two-sided 95% confidence intervals; \(p = 0.950\) is used for one-sided 95% bounds.

Origin and Motivation

The chi-squared distribution arises naturally when assessing the variability of a sample. If \(X_1, X_2, \ldots, X_n\) are independent standard normal variables, the sum of their squares \[Q = \sum_{i=1}^{n} X_i^2\] follows a chi-squared distribution with \(n\) degrees of freedom, written \(Q \sim \chi^2_n\).

In practice, the chi-squared distribution governs the sampling distribution of the sample variance. Specifically, if \(X_1, \ldots, X_n \sim \mathcal{N}(\mu,\,\sigma^2)\), then \[\frac{(n-1)\,s^2}{\sigma^2} \sim \chi^2_{n-1}.\] This relationship is used to construct confidence intervals for \(\sigma^2\) and appears in the Lieberman–Resnikoff tolerance factor (Section [sec:lieberman]).

Definition

The probability density function of the chi-squared distribution with \(\nu\) degrees of freedom is \[f(x) = \frac{1}{2^{\nu/2}\,\Gamma(\nu/2)}\,x^{\nu/2-1}\,e^{-x/2}, \quad x > 0.\] Unlike the normal and \(t\)-distributions, the chi-squared distribution is defined only for \(x > 0\) and is not symmetric: it is right-skewed, with the skewness decreasing as \(\nu\) increases.

Properties

• Mean: \(E[\chi^2_\nu] = \nu\).

• Variance: \(\mathrm{Var}[\chi^2_\nu] = 2\nu\).

• Additivity: If \(Q_1 \sim \chi^2_{\nu_1}\) and \(Q_2 \sim \chi^2_{\nu_2}\) are independent, then \(Q_1 + Q_2 \sim \chi^2_{\nu_1+\nu_2}\).

• Normal approximation: For large \(\nu\), \(\sqrt{2\chi^2_\nu} - \sqrt{2\nu - 1}\) is approximately standard normal (Fisher 1922).

Quantile Table

Table [tab:chi2_quantiles] gives \(\chi^2_{p,\,\nu}\) such that \(P(\chi^2 \leq \chi^2_{p,\,\nu}) = p\). The lower quantiles (small \(p\)) are needed for upper confidence bounds on \(\sigma^2\); the upper quantiles (large \(p\)) are needed for lower bounds. Both appear in tolerance factor calculations.

A single value of \(\bar{x}\) is a point estimate of \(\mu\). A confidence interval provides a range that is expected to contain \(\mu\) with a specified probability.

Known Variance

If \(\sigma\) is known and the observations are normally distributed (or \(n\) is large), the two-sided \((1-\alpha)\) confidence interval for \(\mu\) is \[\bar{x} \pm z_{\alpha/2}\,\frac{\sigma}{\sqrt{n}},\] where \(z_{\alpha/2}\) is the upper \(\alpha/2\) quantile of the standard normal distribution (Table [tab:z_quantiles]). For a 95% interval, \(z_{0.025} = 1.960\).

Unknown Variance (Student’s \(t\))

In practice, \(\sigma\) is unknown and must be estimated by \(s\). The confidence interval then uses Student’s \(t\)-distribution with \(\nu = n - 1\) degrees of freedom: \[\bar{x} \pm t_{\alpha/2,\,n-1}\,\frac{s}{\sqrt{n}}.\] For small \(n\) (typical at the loading bench), the \(t\)-distribution has heavier tails than the normal, producing wider intervals that correctly reflect the additional uncertainty from estimating \(\sigma\). The relevant quantiles are found in Table [tab:t_quantiles].

Practical Example

A handloader fires \(n = 10\) rounds over a chronograph and records a mean velocity \(\bar{x} = 2{,}750\) fps with \(s = 12\) fps. The 95% confidence interval for the true mean velocity is \[2{,}750 \pm 2.262 \times \frac{12}{\sqrt{10}} = 2{,}750 \pm 8.6 \;\text{fps},\] i.e., \([2{,}741.4,\; 2{,}758.6]\) fps. Although the sample mean is 2,750 fps, the true mean could plausibly lie anywhere in this range.

In precision shooting, a common concern is not the average but the worst case: what is the maximum velocity (and hence maximum pressure) that a load might produce? This calls for a one-sided upper confidence bound rather than a two-sided interval.

One-Sided Upper Bound on the Mean

A one-sided \((1-\alpha)\) upper confidence bound for \(\mu\) is \[\mu \leq \bar{x} + t_{\alpha,\,n-1}\,\frac{s}{\sqrt{n}}.\] Note that this uses \(t_{\alpha,\,n-1}\) (not \(t_{\alpha/2}\)), since the entire \(\alpha\) risk is concentrated in one tail.

Upper Tolerance Bound for Individual Values

A bound on the mean tells us about the average round; it does not tell us how fast the hottest round in a large production run might be. For that, we need a tolerance bound: a value that, with confidence \(\gamma\), will exceed at most a fraction \(\beta\) of all individual values.

Under normality, the one-sided upper \((1-\beta)\)-content tolerance bound with confidence \(\gamma\) is \[U = \bar{x} + k(n, \beta, \gamma)\,s,\] where the tolerance factor \(k\) depends on the sample size, the coverage proportion \(\beta\), and the confidence level \(\gamma\).

[Figure: One-sided upper tolerance bound. The shaded blue area covers a proportion \(p\) of the population; the red tail represents the fraction \(1-p\) that may exceed the bound \(\bar{x} + k \cdot s\). this diagram is drawn by LaTeX and appears in the PDF edition.]

The Lieberman–Resnikoff Approximation

Lieberman and Resnikoff (1955) provide a practical approximation for \(k\) that avoids the non-central \(t\)-distribution tables. For a one-sided tolerance bound covering proportion \(\beta\) of the population with confidence \(\gamma\), the factor is approximated by \[k \approx z_\beta\,\sqrt{\frac{n-1}{\chi^2_{1-\gamma,\,n-1}}} + z_\beta^2 \cdot \frac{1}{2n},\] where

• \(z_\beta\) is the standard normal quantile such that \(P(Z \leq z_\beta) = \beta\),

• \(\chi^2_{1-\gamma,\,n-1}\) is the lower \((1-\gamma)\)-quantile of the chi-squared distribution with \(n-1\) degrees of freedom (Table [tab:chi2_quantiles]) — for \(\gamma = 0.95\), the column \(p = 0.050\).

The order of the chi-squared quantile is worth dwelling on, because it is easy to write \(\chi^2_{\gamma}\) out of habit and the result is not merely inaccurate but absurd. An upper tolerance bound must protect against the case where \(s\) has understated the true \(\sigma\); that is the lower tail of \(s^2\), hence the lower quantile of \(\chi^2\), as Section [sec:chi2] already states. Substituting the upper quantile instead gives \(k = 1.34\) for \(n = 5\)—below the asymptotic value \(z_{0.95} = 1.645\), which would mean that five rounds bound the population more tightly than an infinite sample.

A simpler first-order approximation, often sufficient for practical use, is \[k \approx z_\beta\,\sqrt{\frac{n-1}{\chi^2_{1-\gamma,\,n-1}}}.\]

Two features of this table deserve attention. First, the approximation is always low, and a tolerance factor that is too small produces a bound that is too tight—the permissive direction. Where the bound is used for a safety decision, round the exact value up. Second, the deviation does not shrink steadily with \(n\): it is smallest for the very smallest samples (under \(1\,\%\) at \(n = 5\), and under \(0.1\,\%\) at \(n = 4\)), worst around \(n = 20\) to \(50\), and only then improves. The approximation is at its best precisely where one might have expected it to fail.

Note also how large these factors are. At \(n = 5\) the multiplier is \(4.20\)—more than four standard deviations—because \(s\) computed from five rounds is itself a very imprecise estimate of \(\sigma\). A confidence level is not free: demanding \(95\,\%\) rather than \(90\,\%\) raises the factor at \(n = 5\) from \(3.40\) to \(4.20\), a difference of nearly a quarter.

Application: Establishing a Maximum Expected Velocity

Suppose a handloader fires \(n = 10\) rounds and measures \(\bar{x} = 2{,}750\) fps with \(s = 12\) fps. Using the exact \(k = 2.911\) from Table [tab:lr_factors], the one-sided upper bound covering 95% of all rounds with 95% confidence is \[U = 2{,}750 + 2.911 \times 12 = 2{,}784.9 \;\text{fps}.\] The handloader can state with 95% confidence that at least 95% of all rounds from this load will not exceed approximately 2,785 fps. (The Lieberman–Resnikoff approximation would have given 2,784.1 fps—close, but on the optimistic side, as always.) This is directly useful for checking that a load remains within the pressure ceiling specified by the cartridge manufacturer, even accounting for round-to-round variation and the limited sample size.

Conversely, if the maximum safe velocity for a given cartridge is known (from published data or pressure testing), one can work backwards: a load is acceptable only if \(U\) falls below that threshold. This discipline transforms load development from guesswork into an engineering decision bounded by quantified risk.

The Lieberman–Resnikoff approximation (Section [sec:lieberman]) provides a convenient formula for tolerance factors. However, the exact tolerance factors—including those prescribed by the C.I.P. for pressure compliance testing—are derived from the non-central \(t\)-distribution. This section develops the mathematical connection, following the analysis presented by Malfatti (2004).

Problem Statement

Let \(X \sim \mathcal{N}(\mu,\,\sigma^2)\) represent the pressure of a single cartridge, where both \(\mu\) and \(\sigma\) are unknown. From a sample of size \(n\) with sample mean \(\bar{X}\) and sample standard deviation \(s\), we seek the factor \(k\) such that the one-sided tolerance bound \(\bar{X} + k \cdot s\) covers at least a proportion \(p\) of the population with confidence \(\gamma\): \[\mathbb{P}_{\bar{X},\,s}\!\left(\mathbb{P}_{X}\!\left(X \leq \bar{X} + k \cdot s\right) \geq p\right) = \gamma.\]

Reduction to the Non-Central \(t\)

The inner probability condition requires \[\frac{\bar{X} + k \cdot s - \mu}{\sigma} \geq z_p,\] where \(z_p = \Phi^{-1}(p)\) is the standard normal quantile of order \(p\). Rearranging and multiplying both sides by \(\sqrt{n}\): \[k\sqrt{n} \geq \frac{\dfrac{\bar{X} - \mu}{\sigma/\sqrt{n}} + z_p\sqrt{n}}{\dfrac{s}{\sigma}}.\]

We identify two classical random variables in the right-hand side:

• \(Z = \dfrac{\bar{X} - \mu}{\sigma/\sqrt{n}} \sim \mathcal{N}(0,1)\), since \(\bar{X} \sim \mathcal{N}(\mu,\,\sigma^2/n)\).

• \(V = \dfrac{(n-1)\,s^2}{\sigma^2} \sim \chi^2_{n-1}\) (Equation [eq:chi2_variance]).

Writing \(s/\sigma = \sqrt{V/(n-1)}\), the right-hand side of ([eq:ksqrtn]) takes the form \[\frac{Z + \delta}{\sqrt{V/(n-1)}},\] where \(\delta = z_p\sqrt{n}\) is a fixed constant.

Definition of the Non-Central \(t\)-Distribution

If \(Z \sim \mathcal{N}(0,1)\) and \(V \sim \chi^2_\nu\) are independent, the ratio \[T' = \frac{Z + \delta}{\sqrt{V/\nu}}\] follows the non-central \(t\)-distribution with \(\nu\) degrees of freedom and non-centrality parameter \(\delta\), written \(T' \sim t'(\nu,\,\delta)\). When \(\delta = 0\), this reduces to the ordinary (central) Student’s \(t\)-distribution of Section [sec:student]. The non-centrality parameter shifts the distribution away from zero: a positive \(\delta\) shifts the entire distribution to the right.

Solution for the Tolerance Factor

The confidence requirement ([eq:tol_exact]) is satisfied when \(k\sqrt{n}\) equals the \(\gamma\)-quantile of the non-central \(t\)-distribution: \[k = \frac{1}{\sqrt{n}}\;t'_{\gamma,\;\nu,\;\delta},\] where \(t'_{\gamma,\,\nu,\,\delta}\) denotes the value such that \(P(T' \leq t'_{\gamma,\,\nu,\,\delta}) = \gamma\), with \(\nu = n-1\) and \(\delta = z_p\sqrt{n}\).

Verification: C.I.P. Parameters

The C.I.P. standard specifies \(\gamma = 0.95\) (95% confidence) and \(p = 0.99\) (99% population coverage), giving \(z_{0.99} \approx 2.3263\). For a sample of \(n = 5\) rounds:

• Degrees of freedom: \(\nu = 4\).

• Non-centrality parameter: \(\delta = 2.32635 \times \sqrt{5} \approx 5.2018\).

• Non-central \(t\) quantile: \(t'_{0.95,\;4,\;5.2018} = 12.8375\).

Therefore \[k = \frac{12.8375}{\sqrt{5}} = 5.7411,\] which the C.I.P. publishes as \(\mathbf{5.75}\). The remaining factors follow by the same procedure:

The rounding is deliberately upward. For a conformity criterion, an under-rounded \(k\) lowers the bound and admits a lot that should have been rejected; the direction of the rounding error matters more than its magnitude. These are the values used in the safety chapter, and they must agree with it.

The Lieberman–Resnikoff approximation of Section [sec:lieberman] avoids the non-central \(t\) tables by substituting simpler quantities involving only the standard normal and chi-squared distributions. It is accurate to a few percent over the whole useful range, and—contrary to what one might expect—it is at its most accurate for the smallest samples. What it is not is conservative: it falls below the exact factor at every sample size, and a compliance criterion built on it would accept lots that the exact criterion rejects. For safety-critical work such as C.I.P. pressure testing, use the exact factors of Table [tab:nct_factors].

References