body {
font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
background-color: #f8f9fa;
color: #333;
line-height: 1.6;
margin: 0;
padding: 20px;
}
.loan-calc-container {
max-width: 800px;
margin: 20px auto;
background-color: #ffffff;
padding: 30px;
border-radius: 8px;
box-shadow: 0 2px 10px rgba(0, 0, 0, 0.1);
border: 1px solid #e0e0e0;
}
h1, h2 {
color: #004a99;
text-align: center;
margin-bottom: 20px;
}
.input-group {
margin-bottom: 20px;
padding: 15px;
border: 1px solid #dcdcdc;
border-radius: 5px;
background-color: #fefefe;
}
.input-group label {
display: block;
margin-bottom: 8px;
font-weight: bold;
color: #004a99;
}
.input-group input[type="text"],
.input-group input[type="number"] {
width: calc(100% – 20px);
padding: 10px;
border: 1px solid #ccc;
border-radius: 4px;
box-sizing: border-box;
font-size: 16px;
}
button {
display: block;
width: 100%;
padding: 12px 20px;
background-color: #004a99;
color: white;
border: none;
border-radius: 5px;
font-size: 18px;
cursor: pointer;
transition: background-color 0.3s ease;
margin-top: 20px;
}
button:hover {
background-color: #003366;
}
#result {
margin-top: 30px;
padding: 25px;
background-color: #e8f5e9; /* Light success green */
border: 2px dashed #28a745; /* Success green border */
border-radius: 6px;
text-align: center;
}
#result h3 {
margin-top: 0;
color: #28a745;
font-size: 24px;
}
#result-value {
font-size: 36px;
font-weight: bold;
color: #004a99;
}
.article-section {
margin-top: 40px;
padding: 25px;
background-color: #ffffff;
border-radius: 8px;
box-shadow: 0 2px 10px rgba(0, 0, 0, 0.1);
border: 1px solid #e0e0e0;
}
.article-section h2 {
text-align: left;
color: #004a99;
}
.article-section p {
margin-bottom: 15px;
}
.article-section ul, .article-section ol {
margin-left: 20px;
margin-bottom: 15px;
}
.article-section li {
margin-bottom: 8px;
}
.formula {
background-color: #eef5ff;
padding: 10px;
border-left: 3px solid #004a99;
margin: 15px 0;
font-family: Consolas, Monaco, 'Andale Mono', 'Ubuntu Mono', monospace;
font-size: 14px;
overflow-x: auto;
white-space: pre-wrap; /* Allows wrapping long formulas */
}
@media (max-width: 600px) {
.loan-calc-container {
padding: 20px;
}
h1 {
font-size: 24px;
}
#result-value {
font-size: 28px;
}
}
Understanding Standard Deviation
Standard deviation is a fundamental statistical measure that quantifies the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (average) of the set, while a high standard deviation indicates that the values are spread out over a wider range.
In simpler terms, it tells you how consistent or spread out your data is. For example, if a company's monthly sales figures have a low standard deviation, it means their sales are very consistent. If the standard deviation is high, their sales fluctuate significantly from month to month.
Why is Standard Deviation Important?
- Risk Assessment: In finance, a higher standard deviation of investment returns often implies higher risk.
- Quality Control: In manufacturing, a low standard deviation in product measurements indicates consistent quality.
- Data Analysis: It helps in understanding the distribution of data and identifying outliers.
- Predictive Modeling: Understanding data spread is crucial for building accurate predictive models.
How to Calculate Standard Deviation (Step-by-Step)
The process involves several steps:
- Calculate the Mean (Average): Sum all the data points and divide by the number of data points.
- Calculate the Variance: For each data point, subtract the mean and square the result (this is the squared difference). Sum up all these squared differences, and then divide by the number of data points (for population standard deviation) or by the number of data points minus one (for sample standard deviation).
- Calculate the Standard Deviation: Take the square root of the variance.
Formulas:
Let 'x' be each data point, 'μ' (mu) be the population mean, 'σ' (sigma) be the population standard deviation, 'N' be the number of data points in the population, 'x̄' (x-bar) be the sample mean, 's' be the sample standard deviation, and 'n' be the number of data points in the sample.
Population Standard Deviation (σ):
σ = √ [ Σ(xi – μ)² / N ]
Sample Standard Deviation (s):
s = √ [ Σ(xi – x̄)² / (n – 1) ]
This calculator computes the sample standard deviation, which is more commonly used when analyzing a sample of data to infer characteristics about a larger population.
Example Calculation
Let's calculate the standard deviation for the following set of test scores: 75, 80, 85, 90, 95.
1. Calculate the Mean (x̄):
(75 + 80 + 85 + 90 + 95) / 5 = 425 / 5 = 85
2. Calculate Squared Differences from the Mean:
- (75 – 85)² = (-10)² = 100
- (80 – 85)² = (-5)² = 25
- (85 – 85)² = (0)² = 0
- (90 – 85)² = (5)² = 25
- (95 – 85)² = (10)² = 100
3. Sum of Squared Differences:
100 + 25 + 0 + 25 + 100 = 250
4. Calculate the Variance (s²): (Divide by n-1, which is 5-1=4)
250 / 4 = 62.5
5. Calculate the Standard Deviation (s): (Take the square root of the variance)
√62.5 ≈ 7.91
Therefore, the sample standard deviation for these test scores is approximately 7.91.
function calculateStandardDeviation() {
var dataPointsInput = document.getElementById("dataPoints").value;
var resultValueDiv = document.getElementById("result-value");
var resultDescriptionDiv = document.getElementById("result-description");
if (!dataPointsInput) {
resultValueDiv.innerText = "N/A";
resultDescriptionDiv.innerText = "Please enter data points.";
return;
}
var dataPoints = dataPointsInput.split(',')
.map(function(item) {
return parseFloat(item.trim());
})
.filter(function(item) {
return !isNaN(item);
});
if (dataPoints.length < 2) {
resultValueDiv.innerText = "N/A";
resultDescriptionDiv.innerText = "Need at least two valid data points.";
return;
}
var n = dataPoints.length;
var sum = 0;
for (var i = 0; i < n; i++) {
sum += dataPoints[i];
}
var mean = sum / n;
var sumSquaredDifferences = 0;
for (var i = 0; i < n; i++) {
var difference = dataPoints[i] – mean;
sumSquaredDifferences += difference * difference;
}
// Calculate sample variance (divide by n-1)
var variance = sumSquaredDifferences / (n – 1);
// Calculate standard deviation
var standardDeviation = Math.sqrt(variance);
resultValueDiv.innerText = standardDeviation.toFixed(4); // Display with 4 decimal places
resultDescriptionDiv.innerText = "This is the sample standard deviation, measuring the spread of your data around the mean.";
}