Data presentation and interpretation

Explanation

• In a histogram, frequency is proportional to bar area and frequency density is $\frac{\text{frequency}}{\text{class width}}$; unequal class widths make bar height alone misleading.
• Frequency polygons show class patterns, box plots summarise centre and spread, and cumulative frequency diagrams support estimates of medians, quartiles and counts below a value.
• For a continuous probability density histogram, total area is $1$ and the area above an interval is the probability of an observation in that interval.
• Read class boundaries and axis scales before calculating.
• A common error is to use frequency density as though it were frequency.

Explanation

• A scatter diagram shows paired bivariate data; describe the direction and strength of its association and note any outliers or clusters.
• A regression line estimates the mean response for a given explanatory-variable value and is most defensible within the observed data range.
• Distinct clusters may represent different sections of the population, so one overall correlation or regression line can hide different within-group patterns.
• Correlation measures association, not causation; a lurking variable, reverse causation or coincidence may explain the observed relationship.
• For $y=ax^n$, plot $\log y$ against $\log x$ to get gradient $n$ and intercept $\log a$; for $y=kb^x$, plot $\log y$ against $x$ to get gradient $\log b$ and intercept $\log k$.

Explanation

• The mean uses every value, while the median is resistant to extremes; choose a measure that suits the distribution and context.
• Range and interquartile range measure spread using endpoints or quartiles; standard deviation measures typical spread about the mean using all observations.
• For $n$ data values, use $\sigma=\sqrt{\frac{\sum x^2}{n}-\left(\frac{\sum x}{n}\right)^2}$ unless a different convention is stated.
• When groups are combined, add $n$, $\sum x$ and $\sum x^2$ before recalculating; averaging separate standard deviations is not valid.
• Coding data by $y=(x-a)/b$ simplifies calculation, after which transform the statistics back; grouped-data percentiles require linear interpolation within the relevant class.

Explanation

• A common outlier rule flags values below $Q_1-1.5\operatorname{IQR}$ or above $Q_3+1.5\operatorname{IQR}$, but context should guide the final decision.
• Investigate a suspicious value against the original record before correcting or removing it; an unusual valid observation is not automatically an error.
• Handle missing data transparently: record how many values are missing, avoid inventing unsupported values and consider whether missingness could bias conclusions.
• Choose displays to suit the data and purpose: histograms for continuous grouped data, box plots for comparing distributions, and scatter diagrams for paired variables.