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Statistics
Descriptive Statistics
Center, spread, and shape of data
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Descriptive statistics turns raw data into clear summaries that are easier to understand and compare. Instead of staring at a long list of numbers, you can describe the center, spread, and shape of a dataset. These tools matter in science, business, sports, medicine, and everyday decision making because they reveal patterns quickly. A good summary helps you notice what is typical, what varies, and what might be unusual.
Key Facts
- Mean: x̄ = (sum of all data values) / n
- Median: the middle value after sorting the data from least to greatest
- Range: range = maximum value - minimum value
- Sample variance: s^2 = Σ(x - x̄)^2 / (n - 1)
- Sample standard deviation: s = sqrt(Σ(x - x̄)^2 / (n - 1))
- Interquartile range: IQR = Q3 - Q1
Vocabulary
- Mean
- The mean is the arithmetic average found by adding all values and dividing by the number of values.
- Median
- The median is the middle value of an ordered dataset, or the average of the two middle values when there is an even number of values.
- Mode
- The mode is the value or values that occur most often in a dataset.
- Standard deviation
- Standard deviation measures how far data values typically are from the mean.
- Outlier
- An outlier is a data value that is much higher or lower than most of the other values in the dataset.
Common Mistakes to Avoid
- Using the mean when outliers strongly affect the data. The mean can be pulled toward extreme values, so the median may better describe a skewed dataset.
- Forgetting to sort the data before finding the median or quartiles. Median and quartile positions depend on the ordered list, not the original order of collection.
- Confusing range with standard deviation. Range uses only the maximum and minimum, while standard deviation uses every data value and describes typical spread around the mean.
- Treating a graph shape as proof of cause and effect. Descriptive statistics can show patterns and associations, but they do not by themselves prove what caused the pattern.
Practice Questions
- 1 Find the mean, median, mode, and range of the dataset: 4, 7, 7, 10, 12.
- 2 A sample dataset has values 2, 4, 6, 8. Calculate the sample variance and sample standard deviation.
- 3 A dataset of home prices has one extremely expensive mansion compared with many typical homes. Explain whether the mean or median is the better measure of center and why.