What does Population mean in Math?
A population is a complete group of individuals or items that can be distinguished by at least one characteristic mainly for the purpose of data collection and analysis.
Populations are majorly used in statistics and other related branches of mathematics. A small, manageable part of a population is called a sample. Analyzing a sample of a population gives us an idea or information on the complete population.
How do we Compare Populations?
The statistical measures used to compare populations depend upon the type of distributions. A distribution can either be symmetric or skewed.
1. Symmetric Distributions
When both the populations are symmetric, we can compare them using the mean and mean absolute deviation (MAD) measures.
2. Skewed Distributions
If one or both the populations are skewed, we use the median and the interquartile range (IQR) as measures to compare them.
Solved Examples
Example 1: The populations of attendance at basketball and soccer games at St. Patrick’s school is symmetric. John came to a conclusion based on the calculations shown as follows. Is John correct? Explain your answer.
Solution:
From the image,
Attendance for the soccer game:
Mean = 70
MAD = 10
Attendance for the basketball game:
Mean = 150
MAD = 10
John concludes that the difference in the mean of the attendance for both the games is 8 times the MAD, so the attendance at basketball games is significantly greater than the attendance at soccer games.
Since the MAD for the games is ‘high’, it shows the mean value of 150 is not an accurate representation of the other values in the data set.
Hence, John is wrong.
Example 2: The below table illustrates the population (in hundreds) of 4 cities over the past two years. Which of the following cities had the maximum population growth?
Solution:
To check which city had a maximum population growth, let’s subtract the population of all the four cities over the past two years.
The population growth in city A = 3200 – 2900 = 300
The population growth in city B = 7500 – 6400 = 1100
The population growth in city C = 9200 – 8300 = 900
The population growth in city D = 6300 – 4600 = 1700
As a result, city D had the maximum population growth.
Example 3: The double box-and-whisker plot displays the average gas mileage for a sample of cars and buses. Which vehicle gets the better gas mileage?
Solution:
Let’s compare both the populations.
From the plot, we can say that for buses:
First quartile = 17
Median = 19
Third quartile = 23
Similarly, for cars:
First quartile = 23
Median = 27
Third quartile = 30
Buses: 23 miles per gallon is in the third quartile. Therefore, 25% of the buses have a fuel efficiency of at least 23 miles per gallon.
Cars: The first quartile has 23 miles per gallon. Therefore, at least 23 miles per gallon is achieved by 75% of the cars.
So, cars have a better gas mileage than buses.
The mean absolute deviation (MAD) of a data set is defined as the average distance between each data value and the mean.
The range of values in the middle of a distribution is known as the interquartile range (IQR).
The first quartile of a data set is denoted by ‘Q1’ and represents the value under which one-fourth of the points in the data set are included. Similarly, the third quartile ‘Q3’ is the value corresponding to three-fourths of the data points.
Population is the term used to describe a large collection of elements with similar properties. On the other hand, a sample is a portion of the population that has been chosen for the study.