# Two-tailed test

A two-tailed test is when we test if a value is ** different from** the assumed mean (µ). Different from means that it can be

**. So, we are looking to both sides of µ and thus to both tails of the density curve.**

*smaller than*or*greater than*µCompared to one-tailed tests, **two-tailed tests have** **critical areas in both tails at the same time**, whereas one-tailed tests have critical area either or to the left or to the right, not both at the same time.

**On this page**hide

## Two-tailed test, key points

- Two-tailed tests are applied for testing
**statistical significance**in hypothesis tests. - Both of the two tails in the density curve have critical areas as opposed to one-tailed tests with a critical area only to either one of the sides (right or left, not both at the same time) .
- If the tested value falls into either of the critical areas, the null hypothesis is rejected.
- The significance level (α) is
**divided with 2**. E.g. a 5% alpha level leaves divided into 2.5% in each tail.

## Application of the two-tailed test

Hypothesis testing is applied to calculate the statistical significance of a claim. It can typically be a new finding that is tested to see if it is “significantly” different from the assumed mean (µ).

The two-tailed test is applied to test if a certain value is **“significantly” different from** µ. Testing if it is ‘*different from*’ implies to test if it is “significantly” greater than or “significantly” less than µ.

If the value tested for falls beyond the upper limit or lower limit, the null hypothesis is rejected, and it is then inferenced as **“statistically significant”**. The critical values are set at the α/2 level and determine the upper and lower limits of the rejection areas:

**Two-tailed test vs One-tailed test**

As opposed to the one-tailed test, the two-tailed test splits the significance level into both tail which leads to a significantly different basis for the statistical inference:

## Two-tailed test example

Say that a production line of tennis balls supplying The International Tennis Federation must produce and deliver all balls with a diameter of **6.58 cm**. A simple random sample of **100 balls** is planned to be taken and, in the case, that the production has a statistical significance **greater than 5%, **production will be put on hold and the production line adjusted.

The procedure for conducting the two-tailed test is the same as for one-tailed tests and following the general procedure for hypothesis testing. The hypotheses and significance level are specified.

### Step 1: Setting the hypotheses

H0: µ = 6.58

HA: µ ≠ 6.58

### Step 2: Setting significance level

The production will be put on hold at a statistical significance greater than 5%, so the significance level (α) = 0.05

** **

### Step 3: Taking the sample

Now, the company takes a simple random sample of 100 balls and the results are:

- Sample mean (x̄): 6.31
- Sample standard deviation (s): 1.59

** **

### Step 4: Calculating test statistics

The Central Limit Theorem says that the greater the sample size the more any probability distribution approximates to the normal distribution and it is usual to see the this applied for n>30. Our n = 100, so our test statistic will be applied with the normal standard table:

The z-score of -1.698 is within the acceptance area as it is does not go below the critical value of -1.96, nor above the 1.96. Looking up in the z-table, a z-score of -1.698 returns a p-value of 0.089.

### Step 5: Conclusion

The z-score falls in non-rejection area, so we fail to reject the null hypothesis. Therefore, based on our sample, we do not find evidence that the assumed mean should be different from 6.58. The p-value of 0.089 tells us that there is approximately 8.9% probability of getting a result at least as extreme as ours assuming that the mean is 6.58.

## Two-tailed test in MS Excel

The functions **=STANDARDIZE**; **=NORM.S.INV** and **=NORM.S.DIST** can be used to support a hypothesis testing in Excel. Take the following example of **quality assurance **in a production line:

Say a production line must pack 60 Danish butter cookies per box. The boxes are accepted for shipping if they have a maximum of 63 and a minimum of 57 cookies. The company takes a sample of 100 boxes to test if there should be support for the null hypothesis that the mean value is 60 units.

The sample returns a mean of 56 with a standard deviation of 28:

**We fail to reject the null hypothesis** considering the following:

- Test statistic (-1.429) > Critical z (-1.96)
- P-value (0.153) > Significance level (0.05)

## Learning statistics

Some of my preferred material for learnings on two-tailed test:

- Statisticsfun (video 4:34): How to calculate one tail and two tail test
- Jbstatistics (video 9:25): One sided or two-sided test?
- Khan Academy (video 6:34): One tailed and two tailed tests
- Investopedia (text): Two-tailed test definition

#### Carsten Grube

Freelance Data Analyst

##### Normal distribution

##### Confidence intervals

##### Simple linear regression, fundamentals

##### Two-sample inference

##### ANOVA & the F-distribution

# 17 Comments

### Submit a Comment

You must be logged in to post a comment.

+34 616 71 29 85

Call me

Spain: Ctra. 404, km 2, 29100 Coín, Malaga

...........

Denmark: c/o Musvitvej 4, 3660 Stenløse

**Drop me a line**

*What are you working on just now? Can I help you, and can you help me? *

**About me**

Learning statistics. Doing statistics. Freelance since 2005. Dane. Living in Spain. With my Spanish wife and two children.

**Connect with me**

**What they say**

20 years in sales, analysis, journalism and startups. See what my customers and partners say about me.

Hello, just wanted to tell you, I liked this post.

It was practical. Keep on posting!

Its not my first time to go to see this web site, i am

browsing this web page dailly and obtain fastidious information from here everyday.

My coder is trying to persuade me to move to .net from PHP.

I have always disliked the idea because of the expenses.

But he’s tryiong none the less. I’ve been using Movable-type on a

variety of websites for about a year and am worried about switching to another platform.

I have heard great things about blogengine.net.

Is there a way I can transfer all my wordpress posts into

it? Any help would be greatly appreciated!

Hi there! I know this is kinda off topic however I’d figured I’d ask.

Would you be interested in exchanging links or maybe

guest authoring a blog post or vice-versa?

My blog addresses a lot of the same subjects as yours and I believe we

could greatly benefit from each other. If you are interested

feel free to send me an email. I look forward to hearing from you!

Wonderful blog by the way!

Acho que o ideal é manter 2 treinos intervalados com 2 treinos contínuos e

leves – isso cai bastante bem numa rotina. https://sites.google.com/view/orlistat-emagrece-bula-preco/in%C3%ADcio

I go to see each day a few sites and websites to read articles, except this webpage

gives quality based content.

What’s up, I want to subscribe for this weblog to take hottest updates, so where can i do it please assist.

Everyone loves what you guys are usually up too. Such clever work and coverage!

Keep up the amazing works guys I’ve added you guys to blogroll.

Thanks for the auspicious writeup. It in reality used to be a enjoyment account it.

Glance complicated to more brought agreeable from you!

However, how could we keep in touch?

Aw, this was an exceptionally good post. Taking a

few minutes and actual effort to create a superb article… but what can I say…

I put things off a whole lot and don’t manage to get anything done.

Very quickly this website will be famous amid all blogging and site-building users,

due to it’s fastidious posts

What’s up to every body, it’s my first go to see of this web

site; this website carries amazing and genuinely good material designed for visitors.

Hi, this weekend is nice in favor of me, since this point in time i am

reading this enormous informative post here

at my residence.

I’m more than happy to discover this web site.

I need to to thank you for your time for this fantastic read!!

I definitely appreciated every little bit of it and i also have you saved to fav

to check out new stuff on your website.

Quality posts is the secret to interest the users to pay a visit the web site, that’s what this website is providing.

Hello there! This is my first visit to your blog! We are

a group of volunteers and starting a new project in a community in the same

niche. Your blog provided us useful information to work

on. You have done a outstanding job!

Yesterday, while I was at work, my sister stole my apple

ipad and tested to see if it can survive a forty foot drop,

just so she can be a youtube sensation. My iPad is now destroyed and she has 83 views.

I know this is completely off topic but I had to share it with

someone!