Mathwords logoMathwords

Statistical Analysis — Definition, Formula & Examples

Statistical analysis is the process of collecting data, organizing it, summarizing it with numbers and graphs, and using those results to answer a question or make a decision.

Statistical analysis is the systematic application of statistical methods to a data set. It has two branches. Descriptive statistics summarizes the observed data using measures of center (mean, median), measures of spread (range, standard deviation), and data displays. Inferential statistics uses a sample to draw conclusions about a larger population, and it states those conclusions with a measured degree of uncertainty.

Key Formula

xˉ=1n∑i=1nxi\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i
Where:
  • xˉ\bar{x} = the sample mean, a common measure of center
  • nn = the number of data values
  • xix_i = the i-th data value

How It Works

Start with a clear statistical question, one that expects variation in the answers, such as "How long do students in my class take to get to school?" Next, collect data and organize it in a frequency table or an ordered list. Then summarize it. Choose a measure of center and a measure of spread, and pick a display that fits the data type, such as a histogram for one variable or a scatterplot for two. Finally, interpret the results in context and note any limits, like a small sample or an outlier.

Worked Example

Problem: Five students record their commute times to school in minutes: 12, 15, 15, 18, 40. Analyze the data and decide which measure of center best describes a typical commute.
Step 1: Find the mean. Add the values and divide by the number of values.
xˉ=12+15+15+18+405=1005=20\bar{x} = \frac{12 + 15 + 15 + 18 + 40}{5} = \frac{100}{5} = 20
Step 2: Find the median. The data are already in order, so the median is the middle (3rd) value.
12, 15, 15, 18, 40⇒median=1512,\ 15,\ \boxed{15},\ 18,\ 40 \quad\Rightarrow\quad \text{median} = 15
Step 3: Find the range to measure spread.
range=40−12=28\text{range} = 40 - 12 = 28
Step 4: Interpret the results. The value 40 is far above the others, so it is an outlier. It pulls the mean up to 20, which is higher than four of the five commute times. The median is not affected by the outlier.
xˉ=20>15=median\bar{x} = 20 > 15 = \text{median}
Answer: Mean = 20 minutes, median = 15 minutes, range = 28 minutes. Because of the outlier (40), the median of 15 minutes better describes a typical commute.

Visualization

Why It Matters

Statistical analysis is the core of AP Statistics, and it appears on the SAT in data-interpretation questions about tables, scatterplots, and margins of error. Outside school, medical researchers use it to decide whether a new drug works better than a placebo, and sports analysts use it to compare player performance.

Common Mistakes

Mistake: Reporting the mean as the typical value even when the data contain outliers or are strongly skewed.
Correction: Check the shape of the data first. If there is an outlier or strong skew, report the median together with the interquartile range. These measures resist extreme values.
Mistake: Concluding that one variable causes another because a scatterplot shows a strong correlation.
Correction: Correlation shows only that two variables move together. A lurking variable may drive both. Only a well-designed randomized experiment can support a cause-and-effect claim.

Related Terms