Measurement, Parameters, and Statistics: From Raw Data to Meaning

Raw data records individual observations. Statistics begins when we ask which smaller summary preserves the feature needed for a decision.

A representative value should capture the relevant center, remain interpretable, avoid unnecessary instability, and support the decision at hand.

A population is the full group of interest and a sample is the observed subset. A parameter describes the population; a statistic is calculated from the sample.

The mean is a balance point, the median is the ordered middle, and the mode is the most frequent value. For 5, 6, 7, 8, 8, 8, 9, 10 they are 7.625, 8, and 8.

Use the mean for an overall level when values are comparable, the median for a typical value under skew or outliers, and the mode for the most common category or value.

Changing 5, 6, 7, 8, 8, 9, 10 into a dataset that also includes 500 pulls the mean to 67.875, while the median remains 8 and the mode remains 8.

A weighted mean multiplies each value by its importance. Portfolio returns 10%, 6%, and 2% with weights 50%, 30%, and 20% produce 7.2%, not the simple mean 6%.

A hub with 2 deliveries averaging 10 minutes and a hub with 8 averaging 30 minutes have combined mean 26 minutes. The unweighted mean of 20 ignores group sizes.

For a fixed count n, a change Delta in the total changes the mean by Delta divided by n. Adding 10 minutes across five deliveries raises the mean by 2 minutes.

The datasets 48, 49, 50, 51, 52 and 10, 30, 50, 70, 90 both have mean 50, but their distances from the mean are very different.

For 20, 30, 40, 50, 60, 70, 80, the mean is 50 and the population variance is 400. The population standard deviation is 20 in the original score units.

Signed deviations from the mean sum to zero. Squaring makes every contribution nonnegative, gives larger deviations more influence, and yields a smooth objective used throughout statistics and machine learning.

Standard deviation sets a distance scale around the mean, but identical mean and standard deviation do not determine distribution shape. The 68-95-99.7 rule applies specifically to a normal model.

Population variance uses N with the known population mean. When the sample mean is estimated from the same n observations, dividing the squared-deviation sum by n minus 1 removes the downward bias under random sampling.

A sample supplies limited evidence. Its mean and variance are statistics that estimate unknown population parameters, so repeated samples need not give the same answer.

The sample mean is chosen to minimize squared distances within its own sample. One degree of freedom is used to fit that center, so averaging by n underestimates population variance; n minus 1 corrects this on average.

Q1, Q2, and Q3 mark the 25th, 50th, and 75th percentiles under a chosen convention. The interquartile range Q3 minus Q1 describes the middle half of the data.

Being near the 90th percentile means about 90% of observations are at or below that position, subject to the percentile convention used. It describes rank, not how much of a test was answered correctly.

A score of 72 out of 80 is 90%. Its percentile depends on other scores: it may rank near the top, middle, or bottom in different groups.

The z-score z = (x minus mu) divided by sigma expresses signed distance from the mean in standard-deviation units. A score 20 above the mean gives z = 4 when sigma is 5 but z = 0.5 when sigma is 40.

Covariance averages paired deviation products: positive values indicate same-direction movement, negative values opposite-direction movement, and values near zero no linear co-movement. Correlation standardizes by both standard deviations.