What is the pooled variance for the following two samples?
sample 1: n=8 and ss=168
sample 2: n=6 and ss=120
The pooled variance for the given samples is 24. Here option A is the correct answer. The pooled variance is a statistical term that refers to the combined variation of two or more samples. To calculate the pooled variance, we first need to calculate the sample variances and then use a formula to combine them. The formula for pooled variance is:
pooled variance = (SS1 + SS2) / (n1 + n2 - 2)
where SS1 and SS2 are the sum of squares for each sample, and n1 and n2 are the sample sizes. Using the given values, we can calculate the sample variances as follows:
Sample 1 variance = SS1 / (n1-1) = 168 / 7 = 24
Sample 2 variance = SS2 / (n2-1) = 120 / 5 = 24
Now we can use the formula to calculate the pooled variance:
Pooled variance = (SS1 + SS21 + n2 - 2) = (168 + 120) / (8 + 6 - 2) = 288 / 12 = 24
Understanding Pooled Variance
Pooled variance is a crucial concept in statistics, particularly when dealing with multiple samples or groups. It allows us to combine variance estimates from different samples into a single, overall measure of variance. This can be helpful when we want to make inferences or comparisons across different groups.
When we have two or more samples, each with its own variance, we can calculate the pooled variance to get a more accurate estimate of the overall variance. This is important in various statistical analyses, such as ANOVA (Analysis of Variance) and some types of t-tests.
The formula for pooled variance takes into account the sample sizes and sum of squares of each sample. By combining these values appropriately, we can arrive at the pooled variance, which reflects the variability of the entire dataset.
In the given scenario with two samples (sample 1 and sample 2), we first calculate the variances for each sample using the sum of squares and sample size. Once we have the sample variances, we plug these values into the pooled variance formula to obtain the final result.
It's important to note that the pooled variance is influenced by both the within-group variance (variance within each sample) and the between-group variance (variance between the samples). By considering both sources of variance, the pooled variance gives us a more comprehensive understanding of the overall variability in the data.
In summary, pooled variance is a powerful tool in statistics for combining variance estimates across multiple samples. It provides a more accurate measure of variability and enables us to make more robust statistical inferences when comparing groups or conducting hypothesis tests.