Statistically Significant Data

Statistical Analysis and Statistically Significant Data

What are statistics, and why do we need them?

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.

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

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

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.

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.

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.

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.

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.

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.

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).

^{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}