The critical value is a term used in statistics to describe a boundary value that separates the acceptance and rejection of a null hypothesis. It is a numerical value used to make decisions about the significance of results from a hypothesis test.

In this article, we’ll study the basic definition of the term “critical value” its types, and their uses in real life. In the example section, we’ll cover the methods of calculating the critical values.

## What is critical value in statistics?

The critical value is a value that is compared to a test statistic in hypothesis testing to determine whether the null hypothesis is to be rejected or not.

## Types of Critical values:

Generally, in basic statistics, two types (T and Z) are discussed but in this section, we’ll cover the basic and advanced types of critical values along with their real-life uses. The types are as follows:

**T-Critical Value****Z-Critical Value****Chi-Square Critical Value****F-Value**

**T-Critical Value:**

A T-Critical value is used in a t-test to determine whether a sample mean is significantly different from a population mean. It represents the boundary between the acceptance and rejection regions for the t-distribution. The T-Critical value depends on the degrees of freedom and the level of significance (usually represented by alpha).

In a two-tailed test, the T-Critical value is the value such that the area under the t-distribution to the left of the T-Critical value is equal to alpha/2 and the area to the right of the T-Critical value is also equal to alpha/2.

**Real-Life Use of T-critical value:**

T-Critical values are commonly used in fields such as psychology, economics, and education to test the significance of differences between groups. For example, in psychology, a t-test may be used to determine whether the average scores of a treatment group are significantly different from those of a control group.

**Z-Critical Value:**

A Z-Critical value is used in a z-test to determine whether a sample mean is significantly different from a population mean. It is used when the sample size is large (n >= 30) and the population standard deviation is known.

The Z-Critical value is based on the standard normal distribution (Z-distribution) and represents the boundary between the acceptance and rejection regions for the Z-distribution. Like the T-Critical value, the Z-Critical value depends on the level of significance (usually represented by alpha).

**Real Life Use Z-Critical value:**

Z-Critical values are commonly used in fields such as **biology**, medicine, and environmental science to test the significance of differences between groups. For example, in biology, a z-test may be used to determine whether the average height of a species of plant is significantly different from the average height of another species of plant.

**Chi-Square Critical Value**:

The Chi-Square Critical value is used in a chi-square goodness of fit test to determine whether observed data fits a theoretical distribution. The Chi-Square Critical value represents the boundary between the acceptance and rejection regions for the chi-square distribution.

The Chi-Square Critical value depends on the degrees of freedom and the level of significance (usually represented by alpha).

**Real Life Use Chi-Square Critical Value:**

Chi-Square Critical values are commonly used in fields such as sociology, psychology, and marketing to test the significance of differences between observed and expected frequencies.

For example, in sociology, a chi-square test may be used to determine whether the distribution of family sizes in a population is significantly different from the expected distribution.

**F-Value:**

The F-Value is used in an F-test to determine whether the variance of two or more populations is equal. It is calculated by dividing the variance of one group by the variance of another group. The F-Value represents the ratio of the variability between two groups to the variability within the groups.

**Real-Life Use F-Value:**

F-Values are commonly used in fields such as biology.

**Examples of critical value: **

In this section, we’ll describe some mathematical examples to understand the method of finding the critical value of different samples.

**Example 1:**

Evaluate the 1-tail and 2-tailed t critical value if α = 0.03 and the degree of freedom is 50

**Solution: **

**Step 1: **If both **α **and degree of freedom are given then you just have to check the table to find out the value.

df | α | df | α | df | α | df | α |

1 | 10.5791 | 16 | 2.024 | 31 | 1.9522 | 46 | 1.9283 |

2 | 3.8965 | 17 | 2.015 | 32 | 1.9499 | 47 | 1.9273 |

3 | 2.9505 | 18 | 2.0071 | 33 | 1.9477 | 48 | 1.9263 |

4 | 2.6008 | 19 | 2 | 34 | 1.9457 | 49 | 1.9253 |

5 | 2.4216 | 20 | 1.9937 | 35 | 1.9438 | 50 | 1.9244 |

6 | 2.3133 | 21 | 1.988 | 36 | 1.942 | ||

7 | 2.2409 | 22 | 1.9829 | 37 | 1.9402 | ||

8 | 2.1892 | 23 | 1.9782 | 38 | 1.9386 | ||

9 | 2.1504 | 24 | 1.974 | 39 | 1.9371 | ||

10 | 2.1202 | 25 | 1.9701 | 40 | 1.9357 | ||

11 | 2.0961 | 26 | 1.9665 | 41 | 1.9343 | ||

12 | 2.0764 | 27 | 1.9632 | 42 | 1.933 | ||

13 | 2.0601 | 28 | 1.9601 | 43 | 1.9317 | ||

14 | 2.0462 | 29 | 1.9573 | 44 | 1.9305 | ||

15 | 2.0343 | 30 | 1.9547 | 45 | 1.9294 |

On 50 degrees the t critical value is **1.9244** for the **one-tailed T-test**.

df | α | df | α | df | α | df | α |

1 | 21.2052 | 16 | 2.3816 | 31 | 2.2746 | 46 | 2.2395 |

2 | 5.6427 | 17 | 2.3681 | 32 | 2.2712 | 47 | 2.238 |

3 | 3.8961 | 18 | 2.3562 | 33 | 2.268 | 48 | 2.2365 |

4 | 3.2976 | 19 | 2.3457 | 34 | 2.265 | 49 | 2.2351 |

5 | 3.0029 | 20 | 2.3362 | 35 | 2.2622 | 50 | 2.2338 |

6 | 2.8289 | 21 | 2.3278 | 36 | 2.2595 | ||

7 | 2.7146 | 22 | 2.3201 | 37 | 2.257 | ||

8 | 2.6338 | 23 | 2.3132 | 38 | 2.2546 | ||

9 | 2.5738 | 24 | 2.3069 | 39 | 2.2524 | ||

10 | 2.5275 | 25 | 2.3011 | 40 | 2.2503 | ||

11 | 2.4907 | 26 | 2.2958 | 41 | 2.2482 | ||

12 | 2.4607 | 27 | 2.2909 | 42 | 2.2463 | ||

13 | 2.4359 | 28 | 2.2864 | 43 | 2.2445 | ||

14 | 2.4149 | 29 | 2.2822 | 44 | 2.2427 | ||

15 | 2.397 | 30 | 2.2783 | 45 | 2.2411 |

On 50 degrees the t critical value is **2.2338** for the **two-tailed T-test**.

Sometimes we don’t have the table along with us to overcome this you can check Criticalvaluecalculator to get accurate answers according to the statistical tables of the entered values.

**Example 2:**

Find out the z critical (left-tailed) value if α = 0.4

**Solution: **

**Step 1: **Division

Divide 0.4 by “2”.

= 0.4 / 2

= 0.20

**Step 2: **Subtraction

Subtract the value of α from 1.

= 1 – 0.20

= 0.8

**Step 3: **Check the Z-table.

The value matches at 0.8 + 0.05.

So, the approximate value is **0.85.**

**Summary: **

In this article, we have discussed the basic introduction to the term “critical value”. We have also learned the basic definition of this term, along with its types and their real-life uses. All types i.e., T critical value, Z critical value, F value, Chi square value are explained briefly.

In the example section, we have tried to cover the method of finding T critical and Z critical values. Hope you have gone through the basic introduction to this statistical term.