T Score Calculator
Calculate Student's $t$-statistic ($t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$), psychometric $T$-score ($50 + 10Z$), degrees of freedom ($df = n-1$), and SEM.
Calculate Student's t-Statistic & Psychometric T-Score
Enter sample mean (x̄), population mean (μ), sample SD (s), and sample size (n).
Calculation Results
Calculated using Student's t-distribution equations: t = \frac{\bar{x} - \mu}{s / \sqrt{n}}, \quad T_{\text{psychometric}} = 50 + 10Z
Quick Summary
The T Score Calculator evaluates Student's $t$-statistic ($t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$), psychometric $T$-scores ($50 + 10Z$), degrees of freedom ($df = n-1$), and SEM.
Formula Explanation
t = \frac{\bar{x} - \mu}{s / \sqrt{n}}
\text{Psychometric T-score} = 50 + 10 \cdot \left(\frac{X - \mu}{s}\right)How It Works
Student's $t$-statistic tests hypothesis sample means when population standard deviation $\sigma$ is unknown. The T Score Calculator computes standard error $\text{SEM} = s / \sqrt{n}$, Student's $t$-value, and psychometric $T$-score ($50+10Z$) used in psychological testing.
Step-by-Step Worked Example
Practical Problem: A sample of $n = 25$ students scores mean $\bar{x} = 108$ with sample SD $s = 15$ against population mean $\mu = 100$. Calculate $t$-statistic and psychometric $T$-score.
- Step 1: Calculate Standard Error (SEM): $\text{SEM} = 15 / \sqrt{25} = 15 / 5 = \mathbf{3.0000}$.
- Step 2: Calculate difference from mean ($\bar{x} - \mu$): $108 - 100 = \mathbf{8.0000}$.
- Step 3: Divide difference by SEM: $t = 8.0 / 3.0 = \mathbf{+2.6667\text{ Student's t-statistic}}$.
- Step 4: Calculate degrees of freedom: $df = 25 - 1 = \mathbf{24}$.
- Step 5: Calculate Psychometric T-Score ($Z = 8/15 = 0.5333$): $T = 50 + 10(0.5333) = \mathbf{55.33}$.
Real-World Calculation Examples
Scenario 1: Sample Test (n = 25, x̄ = 108, s = 15)
Parameters: x̄ = 108, μ = 100, n = 25
Result: t = +2.6667 (Statistically significant at α = 0.05).
Scenario 2: Psychometric MMPI Personality Test Score
Parameters: Z = +1.5
Result: T-score = 65 (Elevated scale score).
Scenario 3: Bone Density DEXA Scan T-Score
Parameters: Bone mineral density comparison
Result: T = -2.6 (Osteoporosis threshold T ≤ -2.5).
Scenario 4: Small Sample Drug Trial (n = 10)
Parameters: n = 10, df = 9
Result: High critical t-threshold requirement.
Key Benefits of Using This Calculator
Dual Statistical & Psychometric T Output
Calculates Student's $t$-statistic and psychometric $T$-score ($50+10Z$).
Degrees of Freedom ($df$) Calculation
Automatically evaluates degrees of freedom $df = n - 1$.
Standard Error of Mean (SEM)
Computes SEM denominator ($s / \sqrt{n}$).
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is a T-score in hypothesis testing?
Student's $t$-score measures the size of the difference between sample mean $\bar{x}$ and hypothesized mean $\mu$ in units of standard error.
What is a psychometric T-score (50 + 10Z)?
In psychological testing, a T-score transforms Z-scores to have mean 50 and standard deviation 10 ($T = 50 + 10Z$), eliminating negative signs.
What is DEXA scan bone density T-score?
A DEXA T-score compares bone density to a healthy 30-year-old adult ($T \ge -1.0$ Normal; $-1.0 < T < -2.5$ Osteopenia; $T \le -2.5$ Osteoporosis).
How does t-score differ from z-score?
Z-score uses known population standard deviation $\sigma$; $t$-score uses sample standard deviation $s$ and accounts for sample size $n$.
What happens to t-score as sample size n increases?
As sample size $n$ increases ($n > 100$), Student's $t$-distribution approaches the standard normal Z-distribution.
What are degrees of freedom ($df$)?
For a single-sample t-test, degrees of freedom equal $df = n - 1$.
How do I calculate t-statistic in Excel?
Use formula =(AVERAGE(range) - mu) / (STDEV.S(range) / SQRT(COUNT(range))).
Who created Student's t-distribution?
William Sealy Gosset created the t-distribution in 1908 under the pseudonym "Student" while working at Guinness Brewery.
What is critical t-value for 95% confidence?
For two-tailed $\alpha = 0.05$ with $df = 24$, critical $t_{0.025} = 2.064$.
Can t-score be negative?
Yes — a negative $t$-statistic indicates that the sample mean is lower than the hypothesized population mean.