T_DIST

Overview

The T_DIST function provides a unified interface to the main methods of the Student’s t distribution, including PDF, CDF, inverse CDF, survival function, and distribution statistics.

Excel provides several t-distribution functions, including T.DIST, T.DIST.2T, T.DIST.RT for the PDF and CDF, and T.INV, T.INV.2T for the inverse CDF (quantile). The Python function in Excel here also supports the survival function, inverse survival, and distribution statistics (mean, median, variance, standard deviation), as well as location and scale parameters, which are not available in native Excel functions.

The Student’s t distribution is widely used in statistics, especially for hypothesis testing and confidence intervals when the sample size is small and the population standard deviation is unknown. The PDF is given by:

f(x, \mathrm{df}, \mathrm{loc}, \mathrm{scale}) = \frac{\Gamma\left(\frac{\mathrm{df}+1}{2}\right)}{\sqrt{\mathrm{df}\pi}\;\Gamma\left(\frac{\mathrm{df}}{2}\right)} \left(1 + \frac{(x-\mathrm{loc})^2}{\mathrm{df}\;\mathrm{scale}^2}\right)^{-\frac{\mathrm{df}+1}{2}}

for $\mathrm{df} > 0$ , $\mathrm{scale} > 0$ .

The t distribution is symmetric and bell-shaped, like the normal distribution, but has heavier tails. The degrees of freedom parameter (df) controls the shape; as df increases, the t distribution approaches the normal distribution.

For more details, see the official SciPy documentation .

This example function is provided as-is without any representation of accuracy.

Usage

To use the function in Excel:


=T_DIST(value, df, [loc], [scale], [method])

value (float, required for pdf, cdf, icdf, sf, isf):
- For pdf, cdf, sf: the value $x$ at which to evaluate the function
- For icdf, isf: the probability $q$ (must be between 0 and 1)
- For mean, median, var, std: skip parameter
df (float, required): Degrees of freedom. Must be $> 0$ .
loc (float, optional, default=0.0): Location parameter.
scale (float, optional, default=1.0): Scale parameter. Must be $> 0$ .
method (string, optional, default=“pdf”): One of pdf, cdf, icdf, sf, isf, mean, median, var, std.

Method	Description	Output
`pdf`	Probability Density Function: $f(x)$ , the likelihood of a specific value $x$ .	Density at $x$
`cdf`	Cumulative Distribution Function: $P(X \leq x)$ , the probability that $X$ is less than or equal to $x$ .	Probability
`icdf`	Inverse CDF (Quantile Function): Returns $x$ such that $P(X \leq x) = q$ for a given probability $q$ .	Value $x$
`sf`	Survival Function: $P(X > x)$ , the probability that $X$ is greater than $x$ .	Probability
`isf`	Inverse Survival Function: Returns $x$ such that $P(X > x) = q$ for a given probability $q$ .	Value $x$
`mean`	Mean (expected value) of the distribution.	Mean value
`median`	Median of the distribution.	Median value
`var`	Variance of the distribution.	Variance
`std`	Standard deviation of the distribution.	Standard deviation

The function returns a single value (float): the result of the requested method, or an error message (string) if the input is invalid.

Examples

Example 1: PDF at x=0

Inputs:

value	df	loc	scale	method
0	5	0	1	pdf

Excel formula:


=T_DIST(0, 5, 0, 1, "pdf")

Expected output:

Result
0.3796067

Example 2: CDF at x=0

Inputs:

value	df	loc	scale	method
0	5	0	1	cdf

Excel formula:


=T_DIST(0, 5, 0, 1, "cdf")

Expected output:

Result
0.5

Example 3: Inverse CDF (Quantile) at q=0.975

Inputs:

value	df	loc	scale	method
0.975	5	0	1	icdf

Excel formula:


=T_DIST(0.975, 5, 0, 1, "icdf")

Expected output:

Result
2.570582

Example 4: Mean of the distribution

Inputs:

value	df	loc	scale	method
	5	0	1	mean

Excel formula:


=T_DIST( , 5, 0, 1, "mean")

Expected output:

Result
0.0

Python Code


from scipy.stats import t as scipy_t
import math
 
def t_dist(value=None, df=1.0, loc=0.0, scale=1.0, method="pdf"):
    """
    Student's t distribution function supporting multiple methods.
 
    Args:
        value: Input value (float), required for methods except 'mean', 'median', 'var', 'std'.
        df: Degrees of freedom (float, >0).
        loc: Location parameter (float, default: 0.0).
        scale: Scale parameter (float, default: 1.0, >0).
        method: Which method to compute (str): 'pdf', 'cdf', 'icdf', 'sf', 'isf', 'mean', 'median', 'var', 'std'. Default is 'pdf'.
 
    Returns:
        Result of the requested method (float or str), or an error message (str) if input is invalid.
    """
    valid_methods = ['pdf', 'cdf', 'icdf', 'sf', 'isf', 'mean', 'median', 'var', 'std']
    if not isinstance(method, str) or method.lower() not in valid_methods:
        return f"Invalid method: {method}. Must be one of {valid_methods}."
    method = method.lower()
    try:
        df = float(df)
        loc = float(loc)
        scale = float(scale)
    except Exception:
        return "Invalid input: df, loc, and scale must be numbers."
    if df <= 0:
        return "Invalid input: df must be > 0."
    if scale <= 0:
        return "Invalid input: scale must be > 0."
    dist = scipy_t(df, loc, scale)
    # Methods that require value
    if method in ['pdf', 'cdf', 'icdf', 'sf', 'isf']:
        if value is None:
            return f"Invalid input: missing required argument 'value' for method '{method}'."
        try:
            value = float(value)
        except Exception:
            return "Invalid input: value must be a number."
        try:
            if method == 'pdf':
                result = dist.pdf(value)
            elif method == 'cdf':
                result = dist.cdf(value)
            elif method == 'sf':
                result = dist.sf(value)
            elif method == 'isf':
                if not (0 <= value <= 1):
                    return "Invalid input: value (probability) must be between 0 and 1 for isf."
                result = dist.isf(value)
            elif method == 'icdf':
                if not (0 <= value <= 1):
                    return "Invalid input: value (probability) must be between 0 and 1 for icdf."
                result = dist.ppf(value)
        except Exception as e:
            return f"scipy.stats.t error: {e}"
        if isinstance(result, float):
            if math.isnan(result):
                return "Result is NaN (not a number)"
            if math.isinf(result):
                return "inf" if result > 0 else "-inf"
        return result
    # Methods that do not require value
    try:
        if method == 'mean':
            result = dist.mean()
        elif method == 'median':
            result = dist.median()
        elif method == 'var':
            result = dist.var()
        elif method == 'std':
            result = dist.std()
    except Exception as e:
        return f"scipy.stats.t error: {e}"
    if isinstance(result, float):
        if math.isnan(result):
            return "Result is NaN (not a number)"
        if math.isinf(result):
            return "inf" if result > 0 else "-inf"
    return result

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T_DIST

Overview

Usage

Examples

Python Code

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