Elements of Statistics Help
Introduction to the Elements of Statistics
In statistics, the term population refers to a particular set of items, objects, phenomena, or people being analyzed. These items, also called elements , can be actual subjects such as people or animals, but they can also be numbers or definable quantities expressed in physical units.
Consistent with the above definitions of variables, some examples of populations are as follows:
- Assigned radio frequencies (in megahertz) of all FM broadcast transmitters in the United States.
- Temperature readings (in degrees Celsius) at hourly intervals last Wednesday at various locations around the city of New York.
- Minimum barometric-pressure levels (in millibars) at the centers of all the hurricanes in recorded history.
- Brightness levels (in candela) of all the light bulbs in offices in Minneapolis.
- Sound-intensity levels (in decibels relative to the threshold of hearing) of all the electric vacuum cleaners in the world.
Sample, Event, and Census
A sample of a population is a subset of that population. It can be a set consisting of only one value, reading, or measurement singled out from a population, or it can be a subset that is identified according to certain characteristics. The physical unit (if any) that defines a sample is always the same as the physical unit that defines the main, or parent, population. A single element of a sample is called an event .
Consistent with the above definitions of variables, some samples are:
- Assigned radio frequencies of FM broadcast stations whose transmitters are located in the state of Ohio.
- Temperature readings at 1:00 P.M. local time last Wednesday at various locations around the city of New York.
- Minimum barometric-pressure levels at the centers of Atlantic hurricanes during the decade 1991–2000.
- Brightness levels of halogen bulbs in offices in Minneapolis.
- Sound-intensity levels of the electric vacuum cleaners used in all the households in Rochester, Minnesota.
When a sample consists of the whole population, it is called a census . When a sample consists of a subset of a population whose elements are chosen at random, it is called a random sample .
A random variable is a discrete or continuous variable whose value cannot be predicted in any given instance. Such a variable is usually defined within a certain range of values, such as 1 through 6 in the case of a thrown die, or from 88 MHz to 108 MHz in the case of an FM broadcast channel.
It is often possible to say, in a given scenario, that some values of a random variable are more likely to turn up than others. In the case of a thrown die, assuming the die is not “weighted,” all of the values 1 through 6 are equally likely to turn up. When considering the FM broadcast channels of public radio stations, it is tempting to suppose (but this would have to be confirmed by observation) that transmissions are made more often at the lower radio-frequency range than at the higher range. Perhaps you have noticed that there is a greater concentration of public radio stations in the 4-MHz-wide sample from 88 MHz to 92 MHz than in, say, the equally wide sample from 100 MHz to 104 MHz.
In order for a variable to be random, the only requirement is that it be impossible to predict its value in any single instance. If you contemplate throwing a die one time, you can’t predict how it will turn up. If you contemplate throwing a dart one time at a map of the United States while wearing a blindfold, you have no way of knowing, in advance, the lowest radio frequency of all the FM broadcast stations in the town nearest the point where the dart will hit.
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