Mean, Mode and Median are measures of central tendency used in statistics.
Average
The average (Mand) is calculated by adding all the values in a data set and dividing by the number of elements in that set.
As the mean is a measure sensitive to sample values, it is more suitable for situations where the data are more or less evenly distributed, that is, values without large discrepancies.
Formula
Being,
Mand: average
x1, x2, x3,..., xno: data values
n: number of dataset elements
Example
Players on a basketball team have the following ages: 28, 27, 19, 23, and 21 years. What is the average age of this team?
Solution
Read too Simple Average and Weighted Average and Geometric Mean.
Fashion
Fashion (MO) represents the most frequent value of a data set, so to define it, it is enough to observe the frequency with which the values appear.
A dataset is called bimodal when it has two modes, that is, two values are more frequent.
Example
In a shoe store for one day, the following shoe numbers were sold: 34, 39, 36, 35, 37, 40, 36, 38, 36, 38 and 41. What is the fashion value of this sample?
Solution
Observing the numbers sold, we noticed that number 36 was the one with the highest frequency (3 pairs), therefore, the mode is equal to:
MO = 36
median
The Median (Md) represents the core value of a dataset. To find the median value it is necessary to place the values in ascending or descending order.
When the number of elements in a set is even, the median is found by the average of the two central values. Thus, these values are added and divided by two.
Examples
1) In a school, the physical education teacher wrote down the height of a group of students. Considering that the measured values were: 1.54 m; 1.67m, 1.50m; 1.65m; 1.75m; 1.69m; 1.60 m; 1.55 m and 1.78 m, what is the value of the median height of the students?
Solution
First we must put the values in order. In this case, we'll put it in ascending order. Thus, the dataset will be:
1,50; 1,54; 1,55; 1,60; 1,65; 1,67; 1,69; 1,75; 1,78
As the set consists of 9 elements, which is an odd number, then the median will be equal to the 5th element, that is:
Md = 1.65 m
2) Calculate the median value of the following data sample: (32, 27, 15, 44, 15, 32).
Solution
First we need to put the data in order, so we have:
15, 15, 27, 32, 32, 44
As this sample is formed by 6 elements, which is an even number, the median will be equal to the average of the central elements, that is:
To learn more, read also:
- Statistic
- Dispersion Measures
- Variance and Standard Deviation
Solved Exercises
1. (BB 2013 – Carlos Chagas Foundation). In the first four business days of a week, the manager of a bank branch served 19, 15, 17 and 21 customers. On the fifth business day of this week this manager attended n customers.
If the average daily number of customers served by this manager in the five working days of this week was 19, the median was
a) 21.
b) 19.
c) 18.
d) 20.
e) 23.
Although we already know the average, we first need to know the number of customers that were served on the fifth business day. Thus:
To find the median we need to put the values in ascending order, so we have: 15, 17, 19, 21, 23. Therefore, the median is 19.
Alternative: b) 19.
2. (ENEM 2010 - Question 175 - Prova Rosa). The table below shows the performance of a football team in the last championship.
The left column shows the number of goals scored and the right column tells you how many games the team has scored that number of goals in.
| Goals scored | Number of Matches |
|---|---|
| 0 | 5 |
| 1 | 3 |
| 2 | 4 |
| 3 | 3 |
| 4 | 2 |
| 5 | 2 |
| 7 | 1 |
If X, Y and Z are, respectively, the mean, median and mode of this distribution, then
a) X = Y b) Z c) Y d) Z d) Z
We need to calculate the mean, the median and the mode. To calculate the average we must add the total number of goals and divide by the number of matches.
The total number of goals will be found by multiplying the number of goals scored by the number of matches, ie:
Total goals = 0.5+1.3+2.4+3.3+4.2+5.2+7.1 = 45
If the total of matches is equal to 20, the average of goals will be equal to:
To find the fashion value, let's check the most frequent goal count. In this case, we note that in 5 matches no goals were scored.
After this result, the matches that had 2 goals were the most frequent (in all, 4 matches). Therefore,
Z = MO = 0
The median will be found by placing the goal numbers in order. As the number of games was equal to 20, which is an even value, we have to calculate the average between the two central values, so we have:
0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 7
With these results, we know that:
X (average) = 2.25
Y (median) = 2
Z (mode) = 0
That is, Z
Alternative: e) Z
See too:
- Types of Graphics
- Standard deviation
- Statistics - Exercises
- Mathematics in Enem
