install.packages("lsr")Installing package into '/cloud/lib/x86_64-pc-linux-gnu-library/4.5'
(as 'lib' is unspecified)library(lsr)2 chapter1: Measures of central tendency
In this chapter we will learn descriptive statistics.
Measures of Central Tendency are statistical values that represent the center or typical value of a data set. They help us understand the overall trend or “average” behavior of the data.
2.1 Mode in R
The mode is the value(s) that appear most frequently in the data set.
In R, the lsr package provides a simple and clean way to calculate the mode.
Step 1: Install and Load the lsr Package
Example:
data <- c(1, 2, 2, 3, 3, 3, 4, 5)
modeOf(data)[1] 3⚠️ Important Note
There is a base R function also named mode() — but it returns the data type (like "numeric", "character") instead of calculating the statistical mode!
Example: Mode in Data Frame
df <- data.frame(
Score = c(1, 2, 2, 3, 4, 3, NA),
Group = c("A", "B", "B", "B", "A", "A", NA),
Age = c(20, 22, 21, 22, 21, 22, 21)
)
sapply(df, modeOf) Score Group Age
[1,] "2" "A" "22"
[2,] "3" "B" "21"The sapply() function is executed on each column of the dataframe.
2.2 Mode in Python
Calculating the Mode in Python Using pandas
Use .mode() to Find the Mode
Exaple 1:
import pandas as pd
data = pd.Series([1, 2, 2, 3, 4, 4, 5])
mod = data.mode()
print("Mode:", mod.tolist())Mode: [2, 4]#or
print (mod)0 2
1 4
dtype: int64Example 2: Create a Sample Data Frame.
Mode for All Columns:
data = {
'A': [1, 2, 2, 3, 4],
'B': [5, 5, 6, 7, 7],
'C': [10, 10, 10, 11, 12]
}
df = pd.DataFrame(data)
mode_values = df.mode()
print(mode_values) A B C
0 2.0 5 10.0
1 NaN 7 NaNExample 3:
Mode for All Columns:
data = {
"Name": ["Alice", "Bob", "Charlie", "Diana", "Eli", "Fiona"],
"Age": [24, 30, 22, 24, 30, 24],
"Gender": ["Female", "Male", "Male", "Female", "Male", "Female"]
}
df = pd.DataFrame(data)
df Name Age Gender
0 Alice 24 Female
1 Bob 30 Male
2 Charlie 22 Male
3 Diana 24 Female
4 Eli 30 Male
5 Fiona 24 FemaleMode of the "Age" column:
df["Age"].mode()0 24
Name: Age, dtype: int642.3 Median in R
The median is the middle value of a dataset when the numbers are sorted in order.
If the number of values is odd, it’s the middle one.
If it’s even, it’s the average of the two middle numbers.
Basic median() function.
x <- c(10, 20, 30, 40, 50)
median(x)[1] 30With missing values (NA) we can use na.rm = TRUE
y <- c(5, 8, NA, 12)
median(y, na.rm =TRUE)[1] 8Using tapply with categorical variables in R
tapply lets you apply a function (like median) to subsets of a vector, defined by a categorical variable (factor).
Age <- c(24, 30, 22, 24, 30, 24)
Gender <- c("F", "M", "M", "F", "M", "F")
tapply(Age, Gender, median) F M
24 30 Explanation:
First argument → numeric vector (
Age)Second argument → categorical variable (
Gender)Third argument → function to apply (
median)
So this gives you the median age for each gender.
Example : DataFrame
data <- data.frame(
Gender = c("f", "f", "m", "m", "f", "m"),
Anxiety = c(12, 15, NA, 20, 18, 25),
Depression = c(30, NA, 28, 35, 40, NA)
)
data Gender Anxiety Depression
1 f 12 30
2 f 15 NA
3 m NA 28
4 m 20 35
5 f 18 40
6 m 25 NAUse sapply , lapply
sapply(data[, c("Anxiety","Depression")], median, na.rm = TRUE) #1 Anxiety Depression
18.0 32.5 #or
sapply(data[sapply(data, is.numeric)], median, na.rm = TRUE) #2 Anxiety Depression
18.0 32.5 #or
lapply(data[, c("Anxiety","Depression")], median, na.rm = TRUE) #3$Anxiety
[1] 18
$Depression
[1] 32.5Use tapply
tapply(data$Anxiety, data$Gender, median, na.rm = TRUE) f m
15.0 22.5 tapply(data$Depression, data$Gender, median, na.rm = TRUE) f m
35.0 31.5 Use aggregate
aggregate(cbind(Anxiety, Depression) ~ Gender, data = data, median,na.rm = TRUE) Gender Anxiety Depression
1 f 15 35
2 m 20 352.4 Median in Python
1.Using the statistics library
import statistics
data = [10, 12, 15, 17, 18]
median_value = statistics.median(data)
print("Median:", median_value)Median: 152.Using the NumPy library
import numpy as np
data = [10, 12, 15, 17, 18]
median_value = np.median(data)
print("Median:", median_value)Median: 15.03.Using the Pandaslibrary to DataFrame
import pandas as pd
data = {
"Age": [24, 30, 22, 24, 30, 24],
"Score": [85, 90, 78, 92, 88, 95]
}
df = pd.DataFrame(data)
print("Median Age:", df["Age"].median())Median Age: 24.0print("Median Score:", df["Score"].median())Median Score: 89.0Median in data with missing values (NaN)
Sometimes data contains missing values. Pandas and NumPy can handle them easily:
import numpy as np
import pandas as pd
data = [10, 12, np.nan, 17, 18]
# با NumPy
print("Median with NumPy:", np.nanmedian(data))Median with NumPy: 14.5# با Pandas
s = pd.Series(data)
print("Median with Pandas:", s.median())Median with Pandas: 14.5Example :Data Frame (Numeric + Categorical)
import pandas as pd
import numpy as np
data = {
"Age": [24, 30, 22, 24, 30, 24],
"Score": [85, np.nan, 78, 92, np.nan, 95],
"Gender": ["F", "M", "M", "F", "M", "F"]
}
df = pd.DataFrame(data)
print(df) Age Score Gender
0 24 85.0 F
1 30 NaN M
2 22 78.0 M
3 24 92.0 F
4 30 NaN M
5 24 95.0 FCalculate the median for each column.
medians = df.median(numeric_only=True, skipna=True)
print(medians)Age 24.0
Score 88.5
dtype: float64Calculating the median by gender
group_medians = df.groupby("Gender").median(numeric_only=True)
print(group_medians) Age Score
Gender
F 24.0 92.0
M 30.0 78.02.5 Mean in R
2.5.1 Arithmetic Mean in R
In R, we use the mean() function to calculate the mean.
anxiety_scores <- c(12, 15, 20, 22, 18, 30, 25, 19, 17, 50)
mean(anxiety_scores)[1] 22.8Mean with missing data (NA) in R
If the data contains missing values (NA), you must use the argument na.rm = TRUE
data <- c(5, 10, NA, 20)
mean(data, na.rm = TRUE)[1] 11.66667categorical variables: tapply
The researcher wants to examine the average depression scores of two groups (men and women). To do this, we use tapply.
group <- c("f","f","f","m","m","m")
depression <- c(18, 22, 20, 25, 30, 28)
tapply(depression, group, mean) f m
20.00000 27.66667 Calculating the mean in a Data Frame with missing data(NA)
A researcher collected anxiety and depression scores from psychology students. Some students did not answer some questions, and the data is incomplete.
data <- data.frame(
Gender = c("f", "f", "m", "m", "f", "m"),
Anxiety = c(12, 15, NA, 20, 18, 25),
Depression = c(30, NA, 28, 35, 40, NA)
)
data Gender Anxiety Depression
1 f 12 30
2 f 15 NA
3 m NA 28
4 m 20 35
5 f 18 40
6 m 25 NAUse lapply , sapply
Mean of each column “Anxiety”,”Depression” (regardless of gender)
sapply(data[, c("Anxiety","Depression")], mean, na.rm = TRUE) #1 Anxiety Depression
18.00 33.25 #or
sapply(data[sapply(data, is.numeric)], mean, na.rm = TRUE) #2 Anxiety Depression
18.00 33.25 #or
lapply(data[, c("Anxiety","Depression")], mean, na.rm = TRUE) #3$Anxiety
[1] 18
$Depression
[1] 33.25Use tapply
Mean anxiety and depression scores by gender
tapply(data$Anxiety, data$Gender, mean, na.rm = TRUE) f m
15.0 22.5 tapply(data$Depression, data$Gender, mean, na.rm = TRUE) f m
35.0 31.5 Use aggregate
aggregate(cbind(Anxiety, Depression) ~ Gender, data = data, mean,na.rm = TRUE) Gender Anxiety Depression
1 f 15 35
2 m 20 352.5.2 Geometric Mean in R
This indicator is most often used when the data is a ratio or ratio, such as growth rates, investment returns, or population changes.
Method 1: Using the psych package
install.packages("psych")Installing package into '/cloud/lib/x86_64-pc-linux-gnu-library/4.5'
(as 'lib' is unspecified)library(psych)
numbers <- c(4, 8, 16, 32)
geometric.mean(numbers)[1] 11.31371Method 2: Using a formula (without a package)
numbers <- c(4, 8, 16, 32)
gm <- exp(mean(log(numbers)))
print(gm)[1] 11.31371Application of the geometric mean:
Economics: mean annual growth rate of an investment.
Medical science: mean increase in bacterial population .
Management: mean annual sales growth percentage.
2.5.3 Harmonic Mean in R
Harmonic mean is most often used in situations where the data is a ratio or rate (such as speed, productivity, or growth rate).
Method 1: Using the psych package
install.packages("psych")Installing package into '/cloud/lib/x86_64-pc-linux-gnu-library/4.5'
(as 'lib' is unspecified)library(psych)
numbers <- c(2, 4, 8, 16)
harmonic.mean(numbers)[1] 4.266667Method 2: Using a formula (without a package)
numbers <- c(2, 4, 8, 16)
hm <- length(numbers) / sum(1 / numbers)
print(hm)[1] 4.266667Application of harmonic mean:
Physics: Calculating the average speed when the same distance is traveled at different speeds.
Economics: Average interest rates or investment returns.
Management: Average productivity indicators in an organization.
Important
The data must be positive.
If the data contains zero or negative values, the harmonic mean is not defined.
For data with NA, you must use
na.rm=TRUE
2.5.4 Trimmed Mean in R
The Trimmed Mean is a version of the arithmetic mean in which a percentage of the smallest and largest data values are removed to reduce the effect of outliers.
if the data contains very large or very small values, the arithmetic mean can be misleading. In these situations, we use the trimmed mean.
numbers <- c(2, 3, 4, 5, 100)
mean(numbers)[1] 22.8mean(numbers, trim = 0.2)[1] 4The trim value is chosen between 0 and 0.5.
For example, trim=0.1 means that 10% of the data is removed from the beginning and 10% from the end.
Before calculation, the data is sorted in ascending order.
Application
Social sciences: In questionnaires when some responses are very unusual.
Economics: In calculating indices when there are outliers.
Data analysis: In data that has noise or measurement error.
2.6 Mean in Python
2.6.1 Arithmetic Mean in Python
Method 1: Using statistics.mean
import statistics
numbers = [10, 20, 30, 40, 50]
mean_value = statistics.mean(numbers)
print("Mean:", mean_value)Mean: 30Method 2: Using NumPy
import numpy as np
numbers = [10, 20, 30, 40, 50]
mean_value = np.mean(numbers)
print("Mean:", mean_value)Mean: 30.0If the data is small: use
statistics.meanIf the data is large or a matrix: use
numpy.mean.
Method 3: Using Pandas
Example 1: Calculating the mean of a column
import pandas as pd
data = {
"Name":["Sevda", "MG", "Reza", "RG"],
"Math":[20, 12, 18, 15],
"Statistics": [17, 16, 19, 14]
}
df = pd.DataFrame(data)
mean_math = df["Math"].mean()
print("Math:", mean_math)Math: 16.25Example 2: Calculating the mean of all columns
means = df.mean(numeric_only=True)
print(means)Math 16.25
Statistics 16.50
dtype: float64Example 3: Calculating the mean of each row
df["Individual Mean"] = df.mean(numeric_only=True, axis=1)
print(df) Name Math Statistics Individual Mean
0 Sevda 20 17 18.5
1 MG 12 16 14.0
2 Reza 18 19 18.5
3 RG 15 14 14.5Mean with NaN
- Using Pandas The Pandas library ignores
NaNvalues by default in its statistical functions.
import pandas as pd
import numpy as np
data = {
"Name":["Sevda", "MG", "Reza", "RG"],
"Math":[20, 12, np.nan, 15],
"Statistics": [17, 16, 19, np.nan]
}
df = pd.DataFrame(data)
mean_math = df["Math"].mean()
mean_Statistics = df["Statistics"].mean()
print("Math:", mean_math)Math: 15.666666666666666print("Statistics:", mean_Statistics)Statistics: 17.333333333333332means = df.mean(numeric_only=True)
print(means)Math 15.666667
Statistics 17.333333
dtype: float64- Using
NumPywithnp.nanmean()
import numpy as np
numbers = [10, 20, np.nan, 30, 40]
mean_value = np.nanmean(numbers)
print(" Mean:", mean_value) Mean: 25.0Example: DataFrame (numeric + categorical)
import pandas as pd
import numpy as np
data = {
"Name":["Sevda", "MG", "Reza", "RG", "SG"],
"Class":["A", "B", "C", "A", "C"],
"Math":[12, 18, 15, np.nan, 20],
"physics": [17, 16, 19, 14, np.nan],
"Psychology": [np.nan, 18, 13, 15, 19]
}
df = pd.DataFrame(data)
print(df) Name Class Math physics Psychology
0 Sevda A 12.0 17.0 NaN
1 MG B 18.0 16.0 18.0
2 Reza C 15.0 19.0 13.0
3 RG A NaN 14.0 15.0
4 SG C 20.0 NaN 19.0- Mean of all numeric columns
means = df.mean(numeric_only=True)
print(means)Math 16.25
physics 16.50
Psychology 16.25
dtype: float64- Mean of each row (for each student)
df["Individual Mean"] = df.mean(numeric_only=True, axis=1)
print(df) Name Class Math physics Psychology Individual Mean
0 Sevda A 12.0 17.0 NaN 14.500000
1 MG B 18.0 16.0 18.0 17.333333
2 Reza C 15.0 19.0 13.0 15.666667
3 RG A NaN 14.0 15.0 14.500000
4 SG C 20.0 NaN 19.0 19.500000- Calculate the group mean (based on categorical column)
group_means = df.groupby("Class").mean(numeric_only=True)
print(group_means) Math physics Psychology Individual Mean
Class
A 12.0 15.5 15.0 14.500000
B 18.0 16.0 18.0 17.333333
C 17.5 19.0 16.0 17.5833332.6.2 Geometric Mean in Python
- Using
statistics.geometric_mean(Python 3.8 and later)
import statistics
numbers = [2, 8, 4]
gm = statistics.geometric_mean(numbers)
print("geometric_mean:", gm)geometric_mean: 4.0- Using SciPy :
scipy.stats.gmean
from scipy.stats import gmean
numbers = [2, 8, 4]
gm = gmean(numbers)
print("gmean:", gm)gmean: 4.0If a number is zero: the total geometric mean is zero.
- If a number is negative: the geometric mean is not defined (except in special cases).
2.6.3 Harmonic Mean in Python
- Using
statistics.harmonic_mean(Python 3.6 and later)
import statistics
numbers = [2, 4, 4]
hm = statistics.harmonic_mean(numbers)
print("hm :", hm)hm : 3.0- Using SciPy :
scipy.stats.hmean
from scipy.stats import hmean
numbers = [2, 4, 4]
hm = hmean(numbers)
print("hmean:", hm)hmean: 3.0All data must be positive. If a value is zero or negative, the harmonic mean is not defined.
The main application is in rates (such as average speed over equal distances).
2.6.4 Trimmed Mean in Python
The trimmed mean is the same as the regular mean, except that some of the largest and smallest data (for example, the top 10% and bottom 10%) are removed and then the mean is taken.
- Using SciPy (
scipy.stats.trim_mean)
from scipy.stats import trim_mean
numbers = [1, 2, 2, 3, 4, 100]
tm = trim_mean(numbers, 0.1)
print("tm :", tm)tm : 18.666666666666668trim_mean(data, 0.1) : Remove 10% of the data from the beginning and end
trim_mean(data, 0.2) : Remove 20% of the data
End Reza