Making Sense of Data: Statistical Calculations
Statistical calculations transform raw data into meaningful summaries, visualizations, and predictions. From organizing data into frequency tables and box plots to calculating percentiles and z-scores, these techniques help students describe patterns, compare datasets, and draw conclusions — skills used in science, business, sports analytics, and everyday decision-making.
Components of Statistical Calculations
This section covers essential statistical tools and techniques:
- Frequency Tables & Histograms: Organizing data into groups (bins) to show how often values fall within each range.
- Box Plots (Five-Number Summary): Summarizing data with the minimum, Q1 (25th percentile), median, Q3 (75th percentile), and maximum.
- Percentiles & Quartiles: Percentiles divide data into 100 equal parts; quartiles divide it into 4. The 50th percentile is the median.
- Z-Scores: Measuring how many standard deviations a value is from the mean: z = (x - mean) / standard deviation.
Examples of Statistical Calculations
Frequency Table Examples
- Test scores: 72, 85, 91, 68, 77, 84, 95, 73, 88, 80. Group into bins: 60-69 (1), 70-79 (3), 80-89 (4), 90-99 (2).
- A survey of hours studied per week: 0-2 hours (5 students), 3-5 hours (12 students), 6-8 hours (8 students), 9+ hours (3 students).
- Rolling a die 30 times and recording results in a frequency table helps compare observed frequencies against the expected frequency of 5 per face.
Box Plot Examples
- Data: 2, 5, 7, 8, 12, 14, 18, 20, 25. Minimum = 2, Q1 = 7, median = 12, Q3 = 18, maximum = 25.
- Two classes' test scores are displayed as side-by-side box plots. Class A's box is narrow (small IQR, consistent scores), while Class B's box is wide (large IQR, varied scores).
- A box plot reveals outliers — any value more than 1.5 × IQR below Q1 or above Q3 is flagged. If Q1 = 20 and Q3 = 40, IQR = 20, so any value below -10 or above 70 is an outlier.
Percentile Examples
- A student scores in the 85th percentile on a standardized test, meaning they scored higher than 85% of test-takers.
- In a dataset of 50 values sorted in order, Q1 (25th percentile) is approximately the 13th value and Q3 (75th percentile) is approximately the 38th value.
- A baby at the 60th percentile for weight is heavier than 60% of babies the same age.
Z-Score Examples
- A test has mean 75 and standard deviation 10. A score of 90 has z = (90 - 75)/10 = 1.5, meaning 1.5 standard deviations above the mean.
- A score of 60 on the same test has z = (60 - 75)/10 = -1.5, meaning 1.5 standard deviations below the mean.
- Compare scores from two different tests: Student A scored 85 on a test with mean 80 and SD 5 (z = 1.0). Student B scored 92 on a test with mean 85 and SD 10 (z = 0.7). Student A performed relatively better despite the lower raw score.