# Statistically Significant Data

Have you ever used a statistical test before? It's a type of mathematical test used to analyse data. Using a statistical test helps scientists find out if the data that they have collected proves their theory. Otherwise, it's just a collection of numbers.

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## Statistically Significant Data: Definition

Let's begin with a definition.

Statistically significant data refers to results of an experiment that are likely to be attributed to a specific cause.

If data from an experiment is statistically significant, the result is unlikely to have occurred by chance.

It's important to consider sampling error, probability, and certainty.

• Sampling error: a statistical error that occurs when an analyst selects a sample that does not effectively represent the entire population.

• Probability: the likelihood of an event happening.

Probability vs Certainty

Nothing in life is 100% certain. You could grow wings and gain the ability to fly. Yes, it's highly unlikely – but not impossible. There's no way of knowing for sure.

So in statistics, we use probability, not certainty.

## Statistical Analysis and Statistically Significant Data

What are statistics, and why do we need them?

Statistics are tests used to analyse, interpret, and present numerical data.

Without statistical tests, our data doesn't really mean anything. It's just a collection of numbers. But by analysing our data, we can find associations or differences between our data, which back up our results and help us to understand the natural environment better.

### Hypotheses

When planning an experiment, scientists make two hypotheses – the null hypothesis and the alternative hypothesis.

The null hypothesis (H0) states that there will be no significant relationships or differences in the data.

The alternative hypothesis (H1) states that there will be a significant relationship or difference in the data.

Amy wants to know if babies prefer playing with blue toys over orange toys.

H0: Babies do not prefer playing with blue toys over orange toys.

H1: Babies prefer playing with blue toys over orange toys.

If the experiment shows a statistically significant result, the null hypothesis will be rejected. If it doesn't show a statistically significant result, the null hypothesis will be accepted.

## Testing for Statistical Significance

To check that the results of a statistical test are significant, we need to check the significance level and the degrees of freedom.

### Significance Levels

The significance level (also called the p-value) is the probability of incorrectly rejecting the null hypothesis.

The p-value describes probability, not certainty.

In biology and environmental science, the significance level is 0.05. That means that if there is a less than 5% chance of incorrectly rejecting the null hypothesis, the data is considered statistically significant.

Degrees of freedom = n – 1, where n is the size of the data set.

Using degrees of freedom helps us find critical cut-off values for statistical tests. The more degrees of freedom there are, the larger the critical value.

Critical values are found in tables. You can easily find them online, or in a statistics book.

## Statistically Significant Data: Variance

Variance is a way of measuring the differences between two datasets. It considers the spread of data points within a dataset.

Scientists can test for variance using the F-test. How does it work?

1. Calculate the mean of your data set.

2. Subtract each data point from the mean to find its deviation.

3. Square each deviation to ensure you have a positive number.

4. Find the sum of the squares.

5. Divide the squares by n-1 to find out the variances.

6. Divide the larger variance by the smaller variance to find the calculated F-value.

7. Compare the calculated value to the critical value. If the calculated value is less than the critical value, there is a statistically significant variance.

n is the size of your dataset.

### Variance: Example

A meteorologist wanted to see if there is a significant difference between the wind speed in Hull and the wind speed in Nottingham. She wrote two hypotheses.

• Null Hypothesis: There is no significant difference between the wind speed in Hull and the wind speed in Nottingham.

• Alternative Hypothesis: There is a significant difference between the wind speed in Hull and the wind speed in Nottingham.

Then, she collected monthly averages and used them to calculate the variance.

 Month Hull: Wind Speed (kph) Hull: Deviation Hull: Deviation2 Nottingham: Wind Speed (kph) Nottingham: Deviation Nottingham: Deviation2 January 24.3 -4.2 17.64 21.5 -3.2 10.24 February 23.0 -2.9 8.41 20.7 -2.4 5.76 March 21.5 -1.4 1.96 19.8 -1.5 2.25 April 18.9 1.2 1.44 17.6 0.7 0.39 May 17.7 2.4 5.76 16.8 1.5 2.25 June 16.3 3.8 14.44 15.7 2.6 6.76 July 16.1 4.0 16 15.7 2.6 6.76 August 17.1 3.0 9 16.0 2.3 5.29 September 19.3 0.8 0.64 17.4 0.9 0.81 October 21.4 -1.3 1.69 18.8 -0.5 0.25 November 22.4 -2.3 5.29 19.4 -1.1 1.21 December 23.3 -3.2 10.24 20.4 -2.1 4.41 Mean 20.1 N/A N/A 18.3 N/A N/A Sum N/A N/A 92.51 N/A N/A 46.38

For Hull, the mean wind speed is 20.1 kph. The sum of squared deviations is 92.51.

Variance:

92.51 ÷ (12-1)

92.51 ÷ 11 = 8.41

For Nottingham, the mean wind speed is 18.3 kph. The sum of squared deviations is 46.38.

Variance:

46.38 ÷ (12-1)

46.38 ÷ 11 = 4.22

Calculated F-value = 8.41 ÷ 4.22 = 1.99

Finally, the meteorologist found the critical f-value from a table. She made sure to check the degrees of freedom (in this example, 11) and the significance level (0.05).

For this test, the critical F-value is 2.16.

As the calculated F-value is less than the critical F-value, there is a statistically significant variance between the datasets. The meteorologist rejected the null hypothesis.

Fig. 1 – The monthly wind speeds for the two cities look very similar. Without statistical analysis, it would be hard to know that there was a difference.

## Statistically Significant Data: Correlation

Spearman's Rank Correlation Coefficient is used to test for an association or relationship between two variables. The relationship can be positive or negative.

• Positive relationship: an increase in one variable is associated with an increase in the other

• Negative relationship: an increase in one variable is associated with a decrease in the other

When performing a Spearman's Rank test, it's important to understand that correlation ≠ causation. Just because two things are linked doesn't mean that one causes a change in the other.

Chocolate consumption per capita is correlated with Nobel Prizes per capita. Unfortunately, that doesn't necessarily mean that eating more chocolate makes you smarter!

How does Spearman's Rank work?

1. Rank the data points for both variables.

2. Work out the difference between the ranks.

3. Square the difference in ranks to ensure that you have a positive number.

4. Substitute your data into the equation shown below to find the calculated r-value.

5. Compare the calculated value to the critical value. If the calculated value is equal to or above the critical value, there is a statistically significant variance.

When ranking data, it can be from smallest to largest or largest to smallest. Just make sure that you rank both variables using the same method.

Equation: p = 1 – (6 x ∑ D2) ÷ (n(n2-1))

• D: difference in ranks
• n: number of data points in the set

### Correlation: Example

A zoologist wanted to see if the number of spots on a Dalmatian was related to its weight. He wrote two hypotheses.

• Null Hypothesis: The number of spots on a Dalmatian is not related to its weight.

• Alternative Hypothesis: The number of spots on a Dalmatian is related to its weight.

He weighed ten adult Dalmatians and counted how many spots they had.

 Weight (kg) Spots Rank of Weight Rank of Spots Difference between Ranks Difference2 24.8 113 6 5 1 1 22.2 144 3 8 -5 25 19.3 199 1 10 -9 81 28.9 65 9 2 7 49 26.0 129 7 7 0 0 20.1 78 2 3 -1 1 31.2 145 10 9 1 1 23.5 50 4 1 3 9 24.5 123 5 6 -1 1 26.7 110 8 4 4 16 Sum 184

Then, the zoologist inserted the data into the equation.

p = 1 – (6 × 184) ÷ (12(122-1))

p = 1 – (1104 ÷ 1716)

Calculated p-value = 0.356

Finally, the zoologist found the critical p-value. For this test, the critical p-value was 0.553. As the calculated p-value was less than the critical p-value, there is not a statistically significant correlation between the variables. The zoologist accepted the null hypothesis.

Fig. 2 – Did you know that Dalmatians aren't born with their spots? They start developing at around 14 days old. Source: unsplash.com

I hope that this article has clarified statistically significant data for you. Statistically significant data is a result that is very unlikely to have occurred by chance. To determine if your data is statistically significant, you need to compare your calculated value to the critical value (which is dependent on the significance level and the degrees of freedom).

## Statistically Significant Data - Key takeaways

• Statistically significant data refers to results of an experiment that are likely to be attributed to a specific cause.
• We use statistical tests to find associations or differences in our data. This backs up our results and helps us to understand the natural world better.
• When planning an experiment, we write a null hypothesis and an alternative hypothesis. If the result is statistically significant, the null hypothesis is rejected.
• When testing for significance, we need to use significance levels (usually 0.05) and degrees of freedom (n-1).
• Variance measures the difference between two datasets, taking into consideration the spread of data points. An F-test is used to see if variance is statistically significant.
• A correlation tests for an association or relationship between two variables. A Spearman's Rank test is used to see if the correlation is statistically significant.

1. Aloys Leo Prinz, Chocolate consumption and Noble laureates, Social Sciences & Humanities Open, 2020

2. Harry Dean, Are Dalmatians Born With Spots: Most Don’t Know This, The Puppy Mag, 2022

3. Hill’s, Dalmatian Dog Breed Information and Personality Traits, 2022

4. Weather Spark, Climate and Average Weather Year Round in Hull, 2022

5. Weather Spark, Climate and Average Weather Year Round in Nottingham, 2022

#### Flashcards in Statistically Significant Data 14

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What does statistically significant data mean?

Statistically significant data is a result that is very unlikely to have occurred by chance.

How do you know if data is statistically significant?

Once you have calculated the test value, compare it to the critical value. Its relation to the critical value will determine if your data is statistically significant.

How do you analyse statistical significance?

Statistically significant variance can be analysed using an F-test. Alternatively, a statistically significant correlation can be analysed using a Spearman's Rank test.

What is considered a statistically significant correlation?

If the calculated p-value is equal to or above the critical p-value, the correlation is considered to be statistically significant.

What standard deviation is statistically significant?

If the calculated p-value for standard deviation is less than the critical p-value, the standard deviation will be considered statistically significant.

## Test your knowledge with multiple choice flashcards

What is the significance level used in environmental science?

How do you calculate degrees of freedom?

In a variance test, if the critical F-value < the calculated F-value, what happens?

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