## Difference between Z score vs T score

Z score is a conversion of raw data to a standard score, when the conversion is based on the population mean and population standard deviation. When a full data set is available with us, we can compute the Z score. Z score is the subtraction of the population mean from raw score and then divides the result with population standard deviation. T score is a conversion of raw data to the standard score when the conversion is based on the sample mean and sample standard deviation. When the population data set is not available, then we have to pick up some sample data to compute the sample mean and population standard deviation.

### Z score

Under a normal distribution, where full data is available, it is a distance from the mean. Its formula is as given below,

`Z= (x-μ)/σ`

Where,

X=individual raw data

μ = Population mean

σ = Population standard deviation

### T score

T score is the subtraction of individual standard deviation from individual mean and then divide the result with sample standard deviation whole result multiplied by sample size. Its formula is as given below,

`t = {(- μ)/s}*`

= Sample Mean

μ = Population Mean

s = Sample Standard deviation

n = sample size

Let’s take an example to understand the same in a better manner:

In a paper, 3 subdivisions are there- I, II and III. Let the number of students who answered I will correctly be 25% i.e., 75% are not able to correctly answer it. Similarly, let 10% and 20% by the number of people who had answered section II and III correctly, thus 90% and 80% have found section II and III though. We assume that ability measured by these three items is the same and it is normally distributed,

The score for each student in a class is used to calculate the mean of marks which is equal to 50 and a standard deviation of 10. We can compute Z score with the score of 50 as (50-50) / 10 = 0

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We can interpret that student score is 0 distance (in units of standard deviations) from the mean, so the student has scored average.

If the score is 60, Z score is (60-50)/ 10 = 1

We can interpret that student has scored above average – a distance of 1 standard deviation above the mean.

### Head to Head Comparison between Z score vs T score (Infographics)

Below is the top 9 difference between Z score vs T score

### Key Differences between Z score vs T score

Let us discuss some of the major differences between Z score vs T score

- Z score is the standardization from the population raw data or more than 30 sample data to standard score while T score is standardization from the sample data of less than 30 data to standard score
- Z score ranges from -3 to 3, while the T score ranges from 20 to 80.
- As the data size increases, distribution tends to be Z distribution. Both Z score vs T score distribution is part of a normal distribution, but based on the size they differ from each other
- Practically, Z score is extensively used in stock market data and to check chances of a company going into bankrupt, while t score is extensively used in checking bone mineral density and Fracture risk assessments

### Z score vs T score Comparison Table

Let’s look at the top 9 Comparison between Z score vs T score

Sr. No. |
Points of comparison |
Z Score |
T score |

1 | Standardization of data |
Its standardization from population data | Its standardization from Sample Data |

2 | Data Size |
When Population is known or above 30, one can use Z score | When the population is not known or the sample size is less than 30, the T score is used. |

3 | Mean |
An average is always zero. | An average is always 50. |

4 | Range |
It Ranges from -3 to 3. | It ranges from 20 and 80. |

5 | Standard Deviation |
Its standard deviation is always 1 | Its standard deviation is always 10 |

6 | Derived Result |
The derived result can be negative | The derived result can never be negative |

7 | Preference |
Comparatively less preferable, as supports large data | More preferable as it covers a higher range, but with an increase in size it has its inherent limitation |

8 | Distribution |
Z score is part of Z distribution | T score is part of T distribution |

9 | With the increase in size |
With the increase in size, the Z score tends to be used | With the increase in size, its usefulness reduces. |

### Conclusion

Both Z score vs T score is part of hypothesis testing under the normal distribution. If you have a set of measurement scores on different measures using Z-scores you can tell how the scores are placed in their distributions. Then you can compare them. Standardization of scores is an extensively used procedure in the field of research and planning as they help in comparison of various test scores. Standardizing scores before combining them is helps a researcher to get better and comparable results.

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