6. Statistical Measures – Examples and Their Calculations

Mean – Average Sum of all observations divided by number of observations; central tendency of data. $\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$

The mean is a measure of central tendency that helps us understand the average value of a dataset. To calculate it, we sum all the observations and then divide that sum by the number of observations in the dataset. This gives us a single number representing the 'typical' or 'average' observation, which is valuable for summarization and comparison.

Imagine you are a teacher and you want to know the average score of your students in a math test. If the students scored 70, 80, 90, and 100, you would add these scores together (70 + 80 + 90 + 100 = 340) and then divide by the number of students (4). This gives you an average score of 85. This average helps to quickly understand how the class performed overall.

Standard Deviation – Average amount by which each measurement differs from the mean; measures data spread. $SD = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2}$

Standard deviation quantifies the amount of variation or dispersion in a dataset. A low standard deviation means that the data points tend to be close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range of values. To calculate it, first, we determine how far each observation deviates from the mean, square those deviations (to make them positive), average these squared deviations, and then take the square root of that average.

Think about the heights of two basketball teams. If Team A has heights like 6'4", 6'5", 6'3", and Team B has heights of 6'0", 6'5", 6'8", and 6'1", Team A will have a lower standard deviation because their heights are more closely clustered around the average height compared to Team B, which has more varied heights. This tells you about the consistency of the players' heights.

Median – Middle value when data is sorted; splits data into two equal halves; less affected by outliers.

The median is another measure of central tendency. To find the median, we first sort the data from smallest to largest and then identify the middle value. If the number of observations is odd, the median is the middle number. If even, it's the average of the two middle numbers. The median is less affected by extreme values (outliers) than the mean, making it a useful measure when we have skewed data.

Consider a family with five income levels: $30,000,$32,000, $34,000,$36,000, and $90,000. The mean income would be affected by the very high$90,000 figure, giving a misleading average. However, the median is $34,000, which better represents the income level of most family members.

Mode – Identify value with highest frequency; useful for categorizing discrete data.

The mode is the value that appears most frequently in a dataset. It can be particularly useful for categorical data where we want to identify the most common category. A dataset may have one mode, more than one mode (bimodal or multimodal), or no mode if all values occur with the same frequency.

Imagine you run a small shop and keep track of which type of candy sells the most. If you sold 10 chocolate bars, 15 gummy bears, and 5 jellybeans, the mode would be gummy bears since it has the highest sales. This information helps you decide what to stock more of in the future.

Range – Difference between maximum and minimum value; indicates data spread. $Range = x_{max} - x_{min}$

The range is a simple measure of variability that indicates the spread of data. It is calculated by subtracting the smallest observation (minimum) from the largest observation (maximum). The range gives a quick insight into how wide the data is spread out; a larger range indicates more variability in the data.

Think of a professional athlete's performance scores over a season. If his highest score is 30 points and his lowest is 10, his range of performance is 20 points. This range helps coaches understand the consistency of the player’s performance throughout the season.

Given a set of strain values from a sensor: | Measurement (μstrain) | 10 | 12 | 11 | 13 | 14 | 12 | 10 | 11 | 15 | 12 | Mean: $\bar{x} = \frac{10 + 12 + 11 + 13 + 14 + 12 + 10 + 11 + 15 + 12}{10} = \frac{120}{10} = 12$ Standard Deviation first calculate deviations: For example, $10 - 12 = -2 \rightarrow (-2)^2 = 4$, Sum squared deviations ≈ 26, $SD = \sqrt{\frac{26}{9}} \approx 1.7$ Median: Sorted data: 10,10,11,11,12,12,12,13,14,15; middle two are 12 and 12, so median = 12. Mode: 12 (appears 3 times, highest frequency). Range: 15 - 10 = 5

This example walks you through the process of calculating each statistical measure with a specific set of strain values recorded by a sensor. We first determine the mean by summing all the measurements and dividing by the number of measurements. Next, we compute the standard deviation by evaluating how much each measurement deviates from the mean, squaring those deviations, and averaging them before taking the square root. The median is found by sorting the numbers and identifying the middle value(s), while the mode identifies which value occurs most frequently, and the range shows the difference between the highest and lowest measures.

Let's compare this to monitoring temperatures in a greenhouse. If you track the temperature daily for ten days and want to analyze those measurements for patterns. By calculating the mean temperature, you would know the average temperature. The standard deviation shows you how much day-to-day temperatures fluctuate. The median helps find the middle temperature if some days were unusually hot or cold, while the mode indicates which temperature occurred most often. The range shows you how varied the temperature was during that period, helping you manage conditions for optimal plant growth.

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