The qnorm() function is **used to calculate quantiles of the standard normal distribution (also known as the Z-distribution).** It accepts a probability (or probabilities) as input and returns the corresponding quantiles from the standard normal distribution.

**Syntax**

```
qnorm(p, mean = 0, sd = 0, lower.tail = TRUE)
```

**Parameters**

**p**: It represents the significance level to be used

**mean:**It is a vector of means.

**sd:**It is a vector of standard deviation.

**lower.tail****=****TRUE:**It is a probability to the left of p in the normal distribution that is returned.

**Example 1: How to Use qnorm() function**

Let’s say you want to find the Z-value for a probability of 0.90 (90% confidence level):

```
# Calculate the Z-value for a 90% confidence level
z_value <- qnorm(0.90)
z_value
```

**Output**

```
[1] 1.281552
```

Let’s generate a complete plot of the standard normal distribution with a vertical red line indicating the Z-value for a 90% confidence level and a label for that Z-value.

```
# Calculate the Z-value for a 90% confidence level
z_value <- qnorm(0.90)
# Create a sequence of x values
x <- seq(-3, 3, by = 0.01)
# Calculate the probability density function (PDF)
# values for the standard normal distribution
pdf_values <- dnorm(x)
# Create a plot
plot(x, pdf_values,
type = "l",
main = "Standard Normal Distribution",
xlab = "Z-value", ylab = "PDF"
)
# Add a vertical line at the Z-value for the 90% confidence level
abline(v = z_value, col = "red", lwd = 2)
# Add a text label to the vertical line
text(z_value + 0.1, max(pdf_values) * 0.8,
labels = round(z_value, 2),
col = "red"
)
```

You can see the complete plot of the standard normal distribution with a vertical red line suggesting the Z-value (1.28) for a 90% confidence level and a label for that Z-value (1.28). The value in the graph is 1.28, but the vertical line overlaps it.

**Example 2: Finding Multiple Z-values for Given Probabilities**

```
# Calculate Z-values for 80%, 90%, and 95% confidence levels
probabilities <- c(0.80, 0.90, 0.95)
z_values <- qnorm(probabilities)
z_values
```

**Output**

```
[1] 1.281552
```

Let’s plot the graph.

```
# Calculate Z-values for 80%, 90%, and 95% confidence levels
probabilities <- c(0.80, 0.90, 0.95)
z_values <- qnorm(probabilities)
# Create a bar plot to display the Z-values
barplot(z_values, names.arg = c("80%", "90%", "95%"),
main = "Z-values for Confidence Levels",
ylab = "Z-values", col = "light blue")
# Add text labels to the bars with adjusted 'pos' for the 95% label
text(1:3, z_values, labels = round(z_values, 2), pos = c(3, 3, 1), col = "red")
```

We created a barplot showing the Z-values for the 80%, 90%, and 95% confidence levels obtained from the qnorm() function.

The text() function labels the bars with the corresponding Z-values, rounded to two decimal places.

**Example 3: Calculating Z-values for Quantiles of a Normal Distribution**

Let’s find the 90th percentile of a normal distribution with a mean of 50 and a standard deviation of 10.

```
# Calculate the Z-value for the 90th percentile of N(50, 10^2)
mean_value <- 50
sd_value <- 10
quantile_value <- qnorm(0.90, mean = mean_value, sd = sd_value)
quantile_value
```

**Output**

```
[1] 1.281552
```

Let’s plot a graph based on the above output.

```
# Calculate the Z-value for the 90th percentile of N(50, 10^2)
mean_value <- 50
sd_value <- 10
quantile_value <- qnorm(0.90, mean = mean_value, sd = sd_value)
# Create a bar plot to display the quantile value
barplot(quantile_value,
names.arg = "90th Percentile",
main = "Z-value for 90th Percentile",
ylab = "Z-value", col = "blue"
)
# Add text label to the bar
text(1, quantile_value,
labels = round(quantile_value, 2), pos = 2, col = "black"
)
```

You can see a barplot with a single bar representing the Z-value for the 90th percentile of the specified normal distribution.

The text() function labels the bar with the Z-value, rounded to two decimal places.

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Krunal Lathiya is a seasoned Computer Science expert with over eight years in the tech industry. He boasts deep knowledge in Data Science and Machine Learning. Versed in Python, JavaScript, PHP, R, and Golang. Skilled in frameworks like Angular and React and platforms such as Node.js. His expertise spans both front-end and back-end development. His proficiency in the Python language stands as a testament to his versatility and commitment to the craft.