Stats Helpers

problem_bank_helpers.stats.shaded_normal_density(q: float | tuple[float, float], /, mean: float = 0, sd: float = 1, rsd: float = 4, lower_tail: bool = True, add_prob: bool = True, add_q: bool = True, add_legend: bool = False, figsize: tuple[float, float] | None = (8, 6), color: Any = 'xkcd:sky blue', x_label: str = 'x', y_label: str = 'f(x; μ,σ)', legend_text: str | None = None, **kwargs) → Figure[source]

Generate a normal distribution plot with optional listed probability calculation.

Parameters:

  • q (float or tuple[float, float]) – If a float, the upper or lower bound of the shaded area. If a tuple of floats, the lower and upper bounds of the shaded area.

  • mean (float) – The mean of the normal distribution. Defaults to 0

  • sd (float) – The standard deviation of the normal distribution. Defaults to 1

  • rsd (float) – The number of standard deviations to plot on either side of the mean. Defaults to 4

  • lower_tail (bool) – Whether the shaded area should represent the lower tail probability \(\operatorname{P}(X \le x)\), or the upper tail probability \(\operatorname{P}(X \ge x)\). Defaults to True

  • add_prob (bool) – Whether to show the probability of the shaded area will be displayed on the plot. Defaults to True

  • add_q (bool) – Whether the value(s) of q should be displayed on the x-axis of the plot. Defaults to True

  • add_legend (bool) – Whether a legend with the mean and standard deviation values will be displayed on the plot. Defaults to False

  • figsize (tuple or tuple[float, float] or None) – The size of the plot in inches. If None, the default matplotlib figure size will be used as this is passed to matplotlib.pyplot.figure(). Defaults to (8, 6)

  • color (Any) – The color of the shaded area as a valid matplotlib color. Defaults to xkcd:sky blue

  • x_label (str) – The label for the x-axis. Defaults to x

  • y_label (str) – The label for the y-axis. Defaults to f(x; μ,σ)

  • legend_text (str or None, Optional) – The text to display in the legend if add_legend is set to true. By default (None), the legend will display the mean and standard deviation values.

  • **kwargs – Additional keyword arguments to pass to matplotlib.pyplot.figure().

Returns:

The generated matplotlib Figure object.

Return type:

matplotlib.figure.Figure

Raises:

  • TypeError – If the input parameters are not of the expected type.

  • ValueError – If the input values are out of the expected range.

Examples

Shading the region \(P(Z \le z)\) where \(Z\sim N(0,1)\) is the standard normal N(0,1)

pbh.stats.shaded_normal_density(-0.2533)
../_images/stats-1.png

Shading the region \(P(Z \ge z)\) where \(Z\sim N(μ,σ)\) is a normal distribution with mean μ and standard deviation σ.

pbh.stats.shaded_normal_density(7.1, 7, 0.1, lower_tail=False, add_legend=True)
../_images/stats-2.png

Shading the region \(P(a \le Z \le b)\) where \(Z\sim N(0,1)\) is the standard normal N(0,1)

pbh.stats.shaded_normal_density((-1.1, 2))
../_images/stats-3.png

References

Based off of an R function written by Dr. Irene Vrbick for making shaded normal density curves.

The R function by Dr. Irene Vrbick was adapted from here.

problem_bank_helpers.stats.shaded_hypothesis_test(critical_value: float, tail: Literal['left', 'right', 'both'], /, mean: float = 0, sd: float = 1, rsd: float = 4, figsize: tuple[float, float] | None = (8, 6), color: Any = 'xkcd:sky blue', x_label: str = 'x', y_label: str = 'Probability Density', legend: str | None = None, **kwargs) → Figure[source]

Generate a normal distribution plot with appropriate tails for a hypothesis test.

Parameters:

  • critical_value (float) – The critical value to plot. If tail is both, -abs(critical_value) is used for the left tail and abs(critical_value) is used for the right tail.

  • tail (left or right or both) – The type of hypothesis test to plot.

  • mean (float) – The mean of the normal distribution. Defaults to 0

  • sd (float) – The standard deviation of the normal distribution. Defaults to 1

  • rsd (float) – The number of standard deviations to plot on either side of the mean. Defaults to 4

  • figsize (tuple or tuple[float, float] or None) – The size of the plot in inches. If None, the default matplotlib figure size will be used as this is passed to matplotlib.pyplot.figure(). Defaults to (8, 6)

  • color (Any) – The color of the shaded area as a valid matplotlib color. Defaults to xkcd:sky blue

  • x_label (str) – The label for the x-axis. Defaults to x

  • y_label (str) – The label for the y-axis. Defaults to Probability Density

  • legend (str or None, Optional) – The text to display in the legend (title) of the plot.

  • **kwargs – Additional keyword arguments to pass to matplotlib.pyplot.figure().

Returns:

The generated matplotlib Figure object.

Return type:

matplotlib.figure.Figure

Raises:

  • TypeError – If the input parameters are not of the expected type.

  • ValueError – If the input values are out of the expected range.

Examples

Left-tailed hypothesis test with a critical value of -1.645 for the standard normal N(0,1)

pbh.stats.shaded_hypothesis_test(-1.645, "left")
../_images/stats-4.png

Right-tailed hypothesis test with a critical value of 1.645 for the standard normal N(0,1)

pbh.stats.shaded_hypothesis_test(1.645, "right")
../_images/stats-5.png

Two-tailed hypothesis test with a critical value of ±1.96 for the standard normal N(0,1)

pbh.stats.shaded_hypothesis_test(1.96, "both")
../_images/stats-6.png