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To find the mean of a data set, you add up all the individual data points and then divide by the number of data points you have. It's the average value of the data and represents the center around which the numbers are distributed. In our example, the data set consists of five numbers: 2, 6, 6, 7, and -1. Adding these numbers gives us a total of 20.

This total sum is then divided by 5, since there are five numbers in the set, to get the mean. The calculation is expressed as: \(\frac{2 + 6 + 6 + 7 - 1}{5} = 4\). As a result, the mean of this data set is 4.

Understanding the mean is crucial as it serves as a benchmark in further statistical calculations, like variance and standard deviation, which require understanding how spread out the data points are around this average.

Data analysis involves examining, cleaning, and modeling data to gain insights and draw conclusions. It's an essential step in statistics to understand patterns or trends within the data. After calculating the mean of our data set, the next step in data analysis is to understand how each data point differs from the mean.

By subtracting the mean from each data point, we calculate the deviations, which measure how much each individual value varies or 'deviates' from the average. For example, subtract 4 from each number in the set:

• 2 - 4 = -2
• 6 - 4 = 2
• 6 - 4 = 2
• 7 - 4 = 3
• -1 - 4 = -5

These deviations help us understand the distribution of data around the mean. Deviations are the foundation for calculating variance and standard deviation, used to quantify the spread of the data.

Variance and standard deviation are fundamental concepts in statistics used to measure the spread or dispersion of a data set.

• Variance provides a measure of how much the data points tend to deviate from the mean on average. To calculate sample variance, we square each deviation (to account for both positive and negative differences alike) and then find the average of these squared deviations, but for a sample variance, we divide by one less than the number of data points. In our example, this is \(\frac{46}{4} = 11.50\).
• Standard deviation, on the other hand, is the square root of variance and gives us an idea of the average distance of each data point from the mean. It is more intuitive because it is in the same units as the original data. Our calculated standard deviation is approximately \(\sqrt{11.50} \approx 3.39\).

Both measures provide insights into the data's variability. A high variance or standard deviation indicates greater spread around the mean, while a low variance signifies that data points are closer to it. These concepts help in predicting trends and making decisions based on data.