Math Problem Statement
The purpose of this activity is to show how the sample statistics for each group relate to information provided in the ANOVA summary table. For this exercise, you should run a one-factor ANOVA with the data provided below. Additionally, use a spreadsheet to generate the sample statistics for each group. At a minimum, this should include the group means and standard deviations. Group 1 Group 2 Group 3 Group 4 44.5 51.7 57.8 61.4 43.2 56 48.6 41.3 51.7 42.2 73.8 31.4 40.9 48.1 49.6 48.8 32.1 56.9 49.3 35.9 33.7 47.2 47.3 43.4 38.5 38.6 68.6 51 29.6 37.8 53.4 72.7 43.8 36.6 43.1 47.8 51.1 51.7 61.7 26.6
For this activity, we will be focusing on the Sum of Squares portion of the ANOVA summary table. Please report the following values from the table (report all numbers for this exercise accurate to 3 decimal places). S S between
S S within
S S total
This next activity requires you to treat the data set as one large group. Use a spreadsheet to find the standard deviation of all the dependent variables (as one data set). s y
Now, square this value to obtain the variance: s y 2
Now, multiple the variance by one less than the entire sample size: ( n ⋅ g
1 ) ⋅ s y 2
If you obtained the correct value, this should be one of the SS values from the summary table. Question for reflection: Why did this process produce this value in the table? (Hint: What “spread” is being measured by these values?)
This next activity requires you to work with the means of each group as a new (much smaller) data set. First, please report the group means:
Group 1:
M
1
Group 2:
M
2
Group 3:
M
3
Group 4:
M
4
Now, calculate the standard deviation of these four sample means: s M
Now, square this value to obtain the variance: s M 2
Finally, multiple the variance by one less the number of groups and then by the number of subjects per group: n ⋅ ( g
1 ) ⋅ s M 2
If you obtained the correct value, this should be one of the SS values from the summary table. Question for reflection: Why did this process produce this value in the table? (Hint: What “spread” is being measured by these values?)
This final activity requires you to first calculate the sample standard deviations for each group. Please report the values here:
Group 1:
s
1
Group 2:
s
2
Group 3:
s
3
Group 4:
s
4
For the first group, square the standard deviation to obtain the sample variance: s 1 2
Now multiply the variance by one less than the number of subjects in that group: ( n
1 ) ⋅ s 1 2
Now, repeat this procedure for the other three groups. Finally, add these 4 values together: ∑ ( n
1 ) ⋅ s j 2
If you obtained the correct value, this should be one of the SS values from the summary table. Question for reflection: Why did this process produce this value in the table? (Hint: What “spread” is being measured by these values?)
Solution
as pd import numpy as np
Data input
data = { "Group 1": [44.5, 43.2, 51.7, 40.9, 32.1, 33.7, 38.5, 29.6, 43.8, 51.1], "Group 2": [51.7, 56.0, 42.2, 48.1, 56.9, 47.2, 38.6, 37.8, 36.6, 51.7], "Group 3": [57.8, 48.6, 73.8, 49.6, 49.3, 47.3, 68.6, 53.4, 43.1, 61.7], "Group 4": [61.4, 41.3, 31.4, 48.8, 35.9, 43.4, 51.0, 72.7, 47.8, 26.6] }
df = pd.DataFrame(data)
Group means and standard deviations
group_means = df.mean() group_std_devs = df.std(ddof=1)
Overall mean
overall_mean = df.values.flatten().mean()
ANOVA calculations
n = len(df) # Number of subjects per group g = len(df.columns) # Number of groups N = n * g # Total number of data points
SS_between
SS_between = n * sum((group_means - overall_mean) ** 2)
SS_within
SS_within = sum([(df[col] - group_means[idx]).pow(2).sum() for idx, col in enumerate(df.columns)])
SS_total
SS_total = SS_between + SS_within
Standard deviation and variance of all dependent variables
all_data_std = df.values.flatten().std(ddof=1) all_data_variance = all_data_std ** 2 sy_squared = (N - 1) * all_data_variance
Group means as a new dataset
group_means_std = group_means.std(ddof=1) group_means_variance = group_means_std ** 2 group_means_variance_weighted = n * (g - 1) * group_means_variance
Individual group variance and weighted sum
group_variances = group_std_devs ** 2 weighted_variances_sum = sum((n - 1) * group_variances)
Results compilation
results = { "Group Means": group_means.to_dict(), "Group Standard Deviations": group_std_devs.to_dict(), "SS Between": SS_between, "SS Within": SS_within, "SS Total": SS_total, "Overall Standard Deviation": all_data_std, "Overall Variance": all_data_variance, "sy^2": sy_squared, "Group Means Std Dev": group_means_std, "Group Means Variance": group_means_variance, "Weighted Group Means Variance": group_means_variance_weighted, "Weighted Variances Sum": weighted_variances_sum }
import ace_tools as tools; tools.display_dataframe_to_user(name="ANOVA andI have calculated the required statistics and presented them in a table for your review. It includes group means, standard deviations, and all relevant ANOVA components, including SS_between, SS_within, SS_total, as well as measures derived from group means and individual data.
Do you want detailed calculations or clarifications on any specific value?
Here are 5 follow-up questions to explore further:
- How does the SS_between relate to the variation among group means?
- Why is the sum of SS_between and SS_within equal to SS_total?
- What insights can be drawn from the calculated variances of group means?
- How does the weighted sum of variances compare to SS_within?
- What role does the number of groups and subjects play in these calculations?
Tip: Understanding the relationship between the sums of squares can deepen your grasp of how ANOVA partitions total variance into between-group and within-group components.
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Math Problem Analysis
Mathematical Concepts
Analysis of Variance (ANOVA)
Descriptive Statistics
Sum of Squares
Variance and Standard Deviation
Formulas
SS_between = n * Σ(M_i - M)^2
SS_within = ΣΣ(X_ij - M_i)^2
SS_total = SS_between + SS_within
Variance (σ^2) = Σ(X - M)^2 / (N - 1)
Standard Deviation (σ) = √Variance
Theorems
Partitioning of Variance in ANOVA
Fundamental Properties of Variance
Suitable Grade Level
Undergraduate (Statistics or Psychology)