Lucas recorded his lunch cost each day for one week in the table below.
Sunday $4.55, Monday $5.25, Tuesday $5.74, Wednesday $4.30, Thursday $4.20, Friday $5.60, Saturday $6.00. Find the mean, standard deviation, and variance of Lucas's lunch expenditures. Round to the nearest thousandth.
To find the mean, we first add up all the expenditures and divide by the number of days:
Mean = (4.55 + 5.25 + 5.74 + 4.3 + 4.2 + 5.6 + 6) / 7 = 5.036
So the mean cost of Lucas's lunch for the week was $5.036.
To find the standard deviation and variance, we'll need to use the following formulas:
σ = sqrt((Σ(xi - μ)^2) / n)
σ^2 = (Σ(xi - μ)^2) / n
Where:
σ is the standard deviation
σ^2 is the variance
Σ is the sum of
xi is the value of each expenditure
μ is the mean
n is the number of expenditures (in this case, n=7)
Using these formulas, we can calculate:
Standard deviation:
σ = sqrt(((4.55 - 5.036)^2 + (5.25 - 5.036)^2 + (5.74 - 5.036)^2 + (4.3 - 5.036)^2 + (4.2 - 5.036)^2 + (5.6 - 5.036)^2 + (6 - 5.036)^2) / 7) = 0.815
So the standard deviation of Lucas's lunch expenditures was $0.815.
Variance:
σ^2 = ((4.55 - 5.036)^2 + (5.25 - 5.036)^2 + (5.74 - 5.036)^2 + (4.3 - 5.036)^2 + (4.2 - 5.036)^2 + (5.6 - 5.036)^2 + (6 - 5.036)^2) / 7 = 0.664
So the variance of Lucas's lunch expenditures was $0.664^2.
Rounded to the nearest thousandth, the mean is $5.036, the standard deviation is $0.815, and the variance is $0.664.
To find the mean, standard deviation, and variance of Lucas's lunch expenditures, we need to follow these steps:
Step 1: Find the mean
To find the mean (average), we add up all the lunch expenditures and divide by the number of days:
Mean = (Sum of all expenditures) / (Number of days)
Sum of all expenditures = $4.55 + $5.25 + $5.74 + $4.30 + $4.20 + $5.60 + $6.00 = $35.64
Number of days = 7
Mean = $35.64 / 7 = $5.091
Therefore, the mean of Lucas's lunch expenditures is $5.091.
Step 2: Find the standard deviation
To find the standard deviation, we need to subtract the mean from each lunch expenditure, square the result, take the average of these squared differences, and then take the square root of that average:
Step 2a: Subtract the mean from each lunch expenditure and square the result
(Sunday's expenditure - Mean)^2 = ($4.55 - $5.091)^2 = $0.0801
(Monday's expenditure - Mean)^2 = ($5.25 - $5.091)^2 = $0.0256
(Tuesday's expenditure - Mean)^2 = ($5.74 - $5.091)^2 = $0.4264
(Wednesday's expenditure - Mean)^2 = ($4.30 - $5.091)^2 = $0.6241
(Thursday's expenditure - Mean)^2 = ($4.20 - $5.091)^2 = $0.7841
(Friday's expenditure - Mean)^2 = ($5.60 - $5.091)^2 = $0.2696
(Saturday's expenditure - Mean)^2 = ($6.00 - $5.091)^2 = $0.8281
Step 2b: Calculate the average of squared differences
Average of squared differences = (Sum of squared differences) / (Number of days)
Sum of squared differences = $0.0801 + $0.0256 + $0.4264 + $0.6241 + $0.7841 + $0.2696 + $0.8281 = $3.038
Standard deviation = sqrt(Average of squared differences) = sqrt($3.038) = $1.743
Therefore, the standard deviation of Lucas's lunch expenditures is $1.743.
Step 3: Find the variance
The variance is simply the square of the standard deviation:
Variance = (Standard deviation)^2 = ($1.743)^2 = $3.038
Therefore, the variance of Lucas's lunch expenditures is $3.038.
To summarize:
Mean = $5.091
Standard deviation = $1.743
Variance = $3.038