Why is a t-score equal to a z-score in a normal distribution?

In statistics, the z-score and t-score are both used to measure how many standard deviations away from the mean something is. For example, if you got an A on your AP Statistics test, your z-score would be 1.96 because the mean GPA on that test was around 2.8, and you got an A, so your performance was 1.96 standard deviations above the average score of the students who took that exam (1σ). But what happens when you take another AP test with a different mean score? Say it’s 3.5 instead of 2.8?

Understanding the Normal Distribution Curve

A z-score tells you how many standard deviations away an individual score is from the mean. On the other hand, a T-Score tells you how many standard deviations away your scores are from the average of all scores. This can be determined using an INVT calculator. Enter any number into the input field and select the button labeled z-scores. The calculator will show you where that number lies within a z-distribution curve. You can also enter two numbers into the input fields and select T-scores instead, showing where those two numbers fall within a T-distribution curve.

The Basics of Z-Scores

A z score is a number that shows how many standard deviations an observation falls above or below the mean. The mean of all values in a data set can be determined by adding up all the observations and dividing them by how many there are. A standard deviation separates close observations from those that are further apart.

To calculate a value’s z-score, we need to know two things:

  • Where it falls on the scale of high and low values
  • How many standard deviations does it fall away from the mean? This second point can be determined using an INVT calculator or Excel’s =STDEV() function.

How to Calculate T and Z Values

T and Z values can be calculated using the INVT calculator. The formula for calculating T values is as follows:

=T(x, m, s)

Z values are calculated using the following formula: =Z(p, q, n) where p=probability of an event occurring with probability q and n is the sample size. To calculate the standard deviation (s), use this formula: s = square root. To calculate the variance (s2), use this equation: s2 = ∑(X – Xbar)2/N, where Xbar is the average of all values in your data set.

Applying This to SAT/ACT Scores

A student’s SAT score can be converted into their percentile rank by taking the number of standard deviations away from the mean and then converting that number into an equivalent t-score. The same can be done for ACT scores. For example, if someone took an SAT and scored below average on it (about 500), they would have been more likely than not to get this score.

This means their percentile rank would be about 50% (1 standard deviation below the mean), which corresponds with a t-score of -2.00 because it is two standard deviations below the mean. On the other hand, someone who got a perfect score on the SAT (about 2400) would be more likely than not to get this score. Their percentile rank would correspond with 99%, or two standard deviations above the mean, corresponding with a t-score of +2.00.

Putting it All Together

A z score is the number of standard deviations from the mean, and a t-score is the variable of standard deviations from the mean. The difference between them comes down to different scales for measurements of variation. A z-score can be used with any scale, but it’s more common to use it with a scale that ranges from 0 (the average) to 1 (a perfect score). With a standard deviation of 1, most people will fall within 3 standard deviations or 2.97 on either side of the average. So if we’re talking about IQ scores, 95% will fall within an IQ range of 85 and 130. On the other hand, someone might measure their weight as they gain or lose weight.

What are z-scores and t-scores?

Z-scores are used when you compare the data from one group of people with the data from another. If you wanted to know if students at University A were taller than those at University B, you could find out by comparing their heights and then converting their height measurements into z-scores. Z-scores are standard or unitless scores because they represent the number of standard deviations away from the mean where your data point is located.

One way to do this would be to subtract your data point’s value from the mean and divide it by its standard deviation. The result will be a positive number, which tells us how many standard deviations away we are on either side of the mean.

When to use the z-test vs. t-test?

A statistical test will be used when the null hypothesis (H0) and alternative hypothesis (HA) are expressed as H0: The difference between the population mean and sample mean is not statistically significant HA: The difference between the population mean and sample mean is statistically significant.

The first step of any statistical test is to determine if there’s an adequate number of data points for the test. This can be determined by looking at the number of scores that are greater than or less than two standard deviations away from the sample mean. If there are no outliers, there will be no need for a z-test because it only considers extreme data points.

Final Words

A T score measures how far away your data point falls from the group mean. Z scores measure the distance between your data point and the population on either side. Your T or Z score will be larger the further you are from the mean. For example, if someone has an intelligence quotient (IQ) of 130 and someone else has an IQ of 70, their t-scores for IQ would be 13.0 (130) vs. 7.5 (70), respectively.

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Marie Booker

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