Analyzing Population Data and Standard Deviation Notes

Distribution Interpretation: Students will develop the ability to interpret various types of data distributions based on measures of central tendency.
Standard Deviation Calculation: Students will learn to calculate and interpret the standard deviation ( $\sigma$ ) of a population data set to determine the dispersion of values.

Scenario: Weights of individual potatoes in a bag were recorded in grams.
Dataset (Raw): $185$ , $143$ , $156$ , $182$ , $201$ , $217$ , $245$ , $182$ , $199$ .
Dataset (Ordered): $143$ , $156$ , $182$ , $182$ , $185$ , $199$ , $201$ , $217$ , $245$ .
Summative Data: * Summation ( $\sum$ ): $1710$ * Number of samples ( $n$ ): $9$
Calculated Statistics: * Mean ( $\bar{x}$ ): $\frac{1710}{9} = 190$ * Median: The middle value in the sorted list is $185$ . * Mode: The most frequently occurring value is $182$ . * Range: The difference between the maximum and minimum values ( $245 - 143 = 102$ ).

Symmetric Distribution: * Relation: The mean and median are approximately equal. * Symmetry: The data points are distributed approximately symmetrically about the mean. * Example Data: $\text{mean} = 13.1$ , $\text{median} = 13.2$ .
Left-Skewed Distribution: * Relation: Identifying this distribution typically reveals that the median is greater than the mean (\text{median} > ext{mean}). * Visual Characteristic: There is less data on the left side of the graph (a longer tail on the left). * Example Data: $\text{mean} = 14.8$ , $\text{median} = 16.7$ .
Right-Skewed Distribution: * Relation: Identifying this distribution typically reveals that the mean is greater than the median (\text{mean} > ext{median}). * Visual Characteristic: There is less data on the right side of the graph (a longer tail on the right). * Example Data: $\text{mean} = 12.7$ , $\text{median} = 11.6$ .

Variance ( $\sigma^2$ ): This represents the distance from the mean for a data point. It is calculated as the average of the squared differences from the mean.
Standard Deviation ( $\sigma$ ): This is a specific measure of how dispersed the data is in relation to the mean. It is the square root of the variance ( $\sigma = \sqrt{\sigma^2}$ ).

Data Collection: A coach recorded times for an $8$ -member track team in a $400$ -meter race.
Data ( $x$ in seconds): $57.1$ , $59.3$ , $54.6$ , $55.2$ , $55.9$ , $54.9$ , $50.3$ , $53.5$ .
Mean Calculation: * $\sum = 440.8$ * $n = 8$ * $\bar{x} = \frac{440.8}{8} = 55.1$
Deviation Analysis ( $x - \bar{x}$ and $(x - \bar{x})^2$ ): * $57.1 - 55.1 = 2 \rightarrow (2)^2 = 4$ * $59.3 - 55.1 = 4.2 \rightarrow (4.2)^2 = 17.64$ * $54.6 - 55.1 = -0.5 \rightarrow (-0.5)^2 = 0.25$ * $55.2 - 55.1 = 0.1 \rightarrow (0.1)^2 = 0.01$ * $55.9 - 55.1 = 0.8 \rightarrow (0.8)^2 = 0.64$ * $54.9 - 55.1 = -0.2 \rightarrow (-0.2)^2 = 0.04$ * $50.3 - 55.1 = -4.8 \rightarrow (-4.8)^2 = 23.04$ * $53.5 - 55.1 = -1.6 \rightarrow (-1.6)^2 = 2.56$
Standard Deviation Computation: * Sum of squares ( $\sum$ ): $48.18$ * Variance ( $\sigma^2$ ): $\frac{48.18}{8} = 6.0225$ * Standard Deviation ( $\sigma$ ): $\sqrt{6.0225} \approx 2.5$
Interpretation: Given the mean of $55.1$ seconds and a standard deviation of approximately $2.5$ seconds, most of the run times were clustered close together between $52.6$ and $57.6$ seconds.

Data Collection: A cafeteria manager tracked daily waffle sales over an $8$ -day period.
Data ( $x$ in waffles): $36$ , $48$ , $44$ , $57$ , $42$ , $40$ , $56$ , $53$ .
Mean Calculation: * $\sum = 376$ * $n = 8$ * $\bar{x} = \frac{376}{8} = 47$
Deviation Analysis ( $x - \bar{x}$ and $(x - \bar{x})^2$ ): * $36 - 47 = -11 \rightarrow (-11)^2 = 121$ * $48 - 47 = 1 \rightarrow (1)^2 = 1$ * $44 - 47 = -3 \rightarrow (-3)^2 = 9$ * $57 - 47 = 10 \rightarrow (10)^2 = 100$ * $42 - 47 = -5 \rightarrow (-5)^2 = 25$ * $40 - 47 = -7 \rightarrow (-7)^2 = 49$ * $56 - 47 = 9 \rightarrow (9)^2 = 81$ * $53 - 47 = 6 \rightarrow (6)^2 = 36$
Standard Deviation Computation: * Sum of squares ( $\sum$ ): $422$ * Variance ( $\sigma^2$ ): $\frac{422}{8} = 52.75$ * Standard Deviation ( $\sigma$ ): $\sqrt{52.75} \approx 7.3$
Interpretation: With a mean sale of $47$ waffles and a standard deviation of approximately $7.3$ , the data indicates that most days saw sales between $39.7$ and $54.3$ waffles.

Learning Finalization: Review of the learning objectives and key ideas regarding central tendency, distribution symmetry/skewness, and the procedural calculation of variance and standard deviation.