The variance between groups is experimental variance, and the variance within groups are error variance, which when added equal the total variance. Explain.

 Excellent observation! You're referring to a fundamental concept in Analysis of Variance (ANOVA). Let's break it down clearly:

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🎯 Statement:

"The variance between groups is experimental (systematic) variance, and the variance within groups is error (residual) variance, which when added equals the total variance."

This relates to ANOVA's decomposition of total variance in the context of comparing group means.

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📊 Real-World Example (Finance-Oriented):

Suppose you're analyzing average customer satisfaction scores (on a 1–10 scale) across three different banks.

You collect sample data from each bank:

• Bank A, Bank B, and Bank C

You're interested in testing:

Are there statistically significant differences in satisfaction scores across these banks?

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🔄 Variance Components in ANOVA

1. Total Variance (SST)

Represents the overall variation in all satisfaction scores, ignoring the group labels (banks).

$$\text{SST}=\sum (Y_{ij}-\bar{Y}{)}^2$$

Where:

• $Y_{ij}$ = value of observation j in group i

• $\bar{Y}$ = grand mean of all scores

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2. Between-Group Variance (SSB) = Experimental/Systematic Variance

Represents the variation between the group means. It captures the effect of the treatment or group (in this case, which bank the score came from).

$$\text{SSB}=\sum n_i({\bar{Y}}_i-\bar{Y}{)}^2$$

• $\bar{Y}_i$ = mean of group i

• $n_i$ = number of observations in group i

✅ This tells us how much variation in Y is due to the groups being different.

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3. Within-Group Variance (SSW) = Error Variance

Represents the variation within each group — i.e., how much individual scores vary within a bank, regardless of group means.

$$\text{SSW}=\sum (Y_{ij}-{\bar{Y}}_i{)}^2$$

✅ This captures random noise, individual differences, or other uncontrolled factors.

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✅ ANOVA Identity:

$$\text{Total Variance (SST)}=\text{Between-Group Variance (SSB)}+\text{Within-Group Variance (SSW)}$$

Or in sum of squares:

$$\text{SST}=\text{SSB}+\text{SSW}$$

📌 Why This Matters

• ANOVA tests whether SSB (systematic variance) is large enough relative to SSW (error variance) to conclude that groups are significantly different.

• This is done using the F-test:

$$F=\frac{MS\mathrm{B/}}{MSW}=\frac{\text{SSB}\mathrm{/}(k-1)/}{\text{SSW}\mathrm{/}(n-k)}$$

Where:

• $MSB$: Mean square between

• $MSW$: Mean square within

• $k$: number of groups

• $n$: total observations

Summary:

Component	What it Measures	Also Called
Between-Group Variance	Differences due to group effects (e.g., banks)	Experimental/Systematic Variance
Within-Group Variance	Differences within groups due to randomness	Error/Residual Variance
Total Variance	Overall variability in all data	SST = SSB + SSW

When between-group variance is significantly larger than within-group variance, ANOVA concludes that the group means are not equal — i.e., the treatment (like bank) has a significant effect.