Population parameters are values that describe a population. A population parameter could be the average income of individuals in a country, the average weight of individuals in a country, or the average number of children individuals in a country have.
There are several reasons as to why an individual’s value for a population parameter does not necessarily match the population parameter itself. Some of these reasons include:
Individuals may not report their true income, weight, or number of children they have.
Some individuals may have higher incomes, weights, or numbers of children than others do. This can create a bias towards the average.
Individuals may belong to groups that are different from the rest of the population. For example, if only wealthy people live in a certain area, then the population’s average income may be higher than it really is because all of the data comes from only those individuals.
Population median
The population median is the middle value of a population when ordered by value or rank. It is a single numerical value that represents the central rank of a population.
Population medians can be calculated for almost any variable, including numerical values (e.g., income, height, weight), categorical variables (e.g., gender, race, ethnicity), and even non-variables (e.g., favorite color, college attended).
The population median is very useful in statistics for several reasons. First, it can be used as a representative value of a population just like the mean can be. Second, it can be used to estimate what the average value of a population is.
Third, it can be used in cases where there is not enough data to calculate the mean or where the data does not follow a normal distribution—thus preventing the use of other statistics like the median absolute deviation (MAD).
Population mode
The population mode is the most common number in a population. For example, the most common gender in a population is typically male.
Population mode is the average of all values in a distribution. In other words, it is the average value of all populations. It is usually represented by the average of the lowest and highest values in a population.
Population mode can be difficult to determine if there are infinite values within a population. In this case, you must find the average of both ends of the spectrum and divide them to get your population mode.
This statistic is used when we are only concerned with the most common number in a distribution and do not need to exactly determine it. It is useful when we are looking for an estimate of what type of number or what type of person has occurred most frequently in a population.
Uses of population parameters
Population parameters are important in determining the size of a population, whether it is growing or shrinking, and what the makeup of that population is.
Population estimates are frequently used in policy making. For example, if the government wants to increase the birth rate, they would need to know how many people were eligible to have children.
They would also need to know how many children were currently in the country as well as how many were expected in the future. This information would then allow them to plan for infrastructure and resources needed for child rearing.
Similarly, if the government wanted to lower the birth rate, they would need to know how many people were eligible to have children and how many children they had already had. They would then be able to plan ways to lower the number of children entering their population.
Population parameters can also be used in marketing. If a company was trying to reach new customers, they could find out what the average age group of their customers was and where they lived. Then, they could focus their advertising on those groups of people.
Sample mean
A sample mean is the average value of a population parameter. A sample mean is calculated by counting the number of values in a sample (n) and then averaging those values (Σx/n).
For example, let’s say a surveyor asked 100 people what their favorite color was and recorded each answer as either red, blue, or green. If 50 people responded with red, then the average response would be red.
As we mentioned before, the population mean is simply the average value of all possible values in a population. Because we are dealing with populations here, this means that there must be an infinite number of values.
Because of this, it can sometimes be more useful to use the sample median instead of the sample mean when performing calculations or using statistics. The median is simply the middle-most value in a set; so how many numbers are in the set? Half! One must be the median.
Sample median
A popular way to estimate population medians is by using sample medians. A sample median is the numerical value that divides a sample population into two equal groups, with half the samples above and half below the numerical value.
Unfortunately, this method can only be used for populations with an even number of members. For example, you could estimate the population median income by taking a sample of individuals and finding their average income, but you could not do this if they had an even number of individuals (0 people) because there would be no one whose income was the average!
Sample medians can be very useful though, and it is easy to find them. You just need to know how many people are in your sample and how many are above and below the average in your sample population.
Sample mode
A sample or a subset of a population is used to make inferences about the entire population. The average height of people in America can be estimated by taking a sample of people and finding the average height of those in the sample.
However, this average height is only an estimate. It may be very close to the actual average height for people in America, but it cannot be said with absolute certainty that it is.
There are two main reasons for this. The first has to do with how the sample was taken. Was the sample taken randomly? If not, then there might be an unintentional bias in who is included in the sample.
The second has to do with how many people were included in the sample. If there were only five people in the average American’s height estimate, then there is a chance that one of them is quite tall or short and that might have an effect on the final number.
Understanding the population parameter
Before discussing how confidence intervals are constructed, it is important to understand what a population parameter is and how we can estimate it using a sample.
A population parameter is the value of a characteristic of a population that we want to know. For example, we may want to know the average weight of all people. Thus, the average weight of all people is the population parameter.
We can estimate this parameter using a sample drawn from that population. We draw a sample of n individuals from the population and then calculate their average weight. This calculated average is the sample mean and it is used in constructing the confidence interval.
A confidence interval estimates either the mean or median of a normal distribution. In this case, we are estimating the population mean.
Estimating the population parameter
A statistical value that represents the entire population is called the population parameter. The population parameter is usually one of three types of values:
Average or mean – often called the average, this is a numerical value that represents what the entire population thinks of as an average value
Median – this is the numerical value that half of the population thinks of as higher and half thinks of as lower; it is not necessarily equivalent to the average
Mode – this is the most common numerical value, no matter what the population thinks of as an average or median
Sometimes, statisticians use more than one sample to estimate a parameter. In these cases, they must compute a new estimate for their parameter based on their new sample(s). This process is known as recalculation.
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