## Measurements and Their Analysis

Sample

Sample

- Subset of data collected from a population

- Can be collected in an economical fashion

- May or may not be representative of a population

- Size of same set influences its ability to predict true values of a population.

- Larger samples generally have a higher probability of predicting true values.

**True Value**

- A quantities theoretically correct or exact value.

- True values are never known

- Often we observe a calibration value with a more precise instrument using very precise techniques. These values are the used as true values with standard equipment.

**Most Probable Value**

- The most likely value for the true value as computed from a sample data set.

- For a simple set of repeated observations, this is the mean (average value).

**Redundancies**

- Also called degrees of freedom

- The number of observations in excess of the number necessary to solve for the unknowns.

How many redundancies are their if a distance is observed three times?

**Methods of Analyzing Data**

- Numerical methods

- Range: The difference between the largest and smallest values in a set of data

- That is, the largest discrepancy in a set of data

- Also called dispersion.

- Discrepancy: The difference between two elements in a set of data.

- Measures of central tendency

- Mean, median, mode

- Measures of data variation

- Variance, standard deviation

- Graphical methods

- Frequency distribution: A bar graph based on a selected class width.

- Also called a histogram

- Class: A sub-region of data

- Class width: range of a single class

- Median: The physical midpoint of a data set when the data is arranged in numerical order

- If the data set has an odd number of elements, then it is the physical midpoint

- If the data set has an even number of elements, then it is the average of the two points nearest to the physical midpoint.

**Creating a Frequency Distribution**

1. Select the number of classes

- Typically between 5 and 10

2. Determine the class width

- Class width = Range/(number of classes)

3. Create a class frequency table

- Determine the number of elements in each class

4. Plot frequency of each class; i.e., create histogram.

**Things To Look For**

- Notice the following when looking at a histogram

- Symmetry about central values

- Range of data set

- Frequency of occurrence of measured values

- Steepness of histogram

- Measure of precision

- The steeper the slopes of the sides are the more precise the data.

- How to Lie with a bar graph

- Note lowest value in histogram

- When comparing two data sets, they must use the same class width