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To calculate the Mean Squared Error (MSE), you first find the forecast errors by subtracting the forecast from the actual sales for each period. Next, you square each of these errors.

Conditions: You have calculated the forecasts for each period.; You have the actual sales data for each period.; Errors are not calculated for the first period if $F_1 = A_1$.

To calculate the Mean Squared Error (MSE), you first find the forecast errors by subtracting the forecast from the actual sales for each period. Next, you square each of these errors.

Conditions: You have calculated the forecasts for each period.; You have the actual sales data for each period.; Errors are not calculated for the first period if $F_1 = A_1$.

In exponential smoothing, the first period's forecast is typically initialized with the first period's actual value ($F_1 = A_1$). Because the forecast and the actual value are identical, the error is zero and uninformative.

Conditions: The initial forecast is set to the first actual value ($F_1 = A_1$).; You are calculating forecast errors to evaluate model accuracy.

In exponential smoothing, the first period's forecast is typically initialized with the first period's actual value ($F_1 = A_1$). Because the forecast and the actual value are identical, the error is zero and uninformative.

Conditions: The initial forecast is set to the first actual value ($F_1 = A_1$).; You are calculating forecast errors to evaluate model accuracy.

The column $(x - \bar{x})(y - \bar{y})$ represents the product of the x-deviation and the y-deviation for each observation. It is used to build the numerator of the least-squares slope formula, capturing how x and y vary together relative to their means.

Conditions: The x-deviations and y-deviations have been computed for each row.; The data set consists of paired observations.

The column $(x - \bar{x})(y - \bar{y})$ represents the product of the x-deviation and the y-deviation for each observation. It is used to build the numerator of the least-squares slope formula, capturing how x and y vary together relative to their means.

Conditions: The x-deviations and y-deviations have been computed for each row.; The data set consists of paired observations.