Understanding Quartile Deviation in Normal Distribution

The question asks for the quartile deviation of a Normal Distribution. The quartile deviation (QD), also known as the Semi-Interquartile Range (SIQR), measures the dispersion or spread of the middle 50% of the data. It is defined as half the difference between the third quartile (\(Q_3\)) and the first quartile (\(Q_1\)).

Mathematically, the formula for quartile deviation is:

\(QD = \frac{Q_3 - Q_1}{2}\)

Quartiles of a Normal Distribution

For a Normal Distribution with mean \(\mu\) and standard deviation \(\sigma\), the quartiles \(Q_1\) and \(Q_3\) are located at specific distances from the mean. These distances are determined by the properties of the normal curve.

• The first quartile (\(Q_1\)) is the value below which 25% of the data falls. For a Normal Distribution, \(Q_1\) is located at approximately 0.6745 standard deviations below the mean. So, \(Q_1 \approx \mu - 0.6745\sigma\).
• The third quartile (\(Q_3\)) is the value below which 75% of the data falls. For a Normal Distribution, \(Q_3\) is located at approximately 0.6745 standard deviations above the mean. So, \(Q_3 \approx \mu + 0.6745\sigma\).

Calculating the Quartile Deviation

Now we can substitute the approximate values of \(Q_1\) and \(Q_3\) for the Normal Distribution into the QD formula:

\(QD = \frac{(\mu + 0.6745\sigma) - (\mu - 0.6745\sigma)}{2}\)

\(QD = \frac{\mu + 0.6745\sigma - \mu + 0.6745\sigma}{2}\)

\(QD = \frac{2 \times 0.6745\sigma}{2}\)

\(QD = 0.6745\sigma\)

So, the quartile deviation of a Normal Distribution is approximately 0.6745 times its standard deviation.

Relating QD to the Options

The given options are simple fractions:

1. \(\frac{1}{2} = 0.5\)
2. \(\frac{2}{3} \approx 0.6667\)
3. \(\frac{1}{3} \approx 0.3333\)
4. \(\frac{1}{4} = 0.25\)

The calculated value of the constant multiplier for the standard deviation, 0.6745, is very close to \(0.6667\), which is \(\frac{2}{3}\). This value, 0.6745, is often approximated as \(\frac{2}{3}\) in this context.

Therefore, the quartile deviation of a Normal Distribution is approximately \(\frac{2}{3}\) times its standard deviation (\(QD \approx \frac{2}{3}\sigma\)). The options are likely asking for the constant fraction that relates the quartile deviation to the standard deviation.

Comparing 0.6745 with the options, the closest value is \(\frac{2}{3}\).

Thus, the quartile deviation of Normal Distribution is approximately \(\frac{2}{3}\) times the standard deviation, and the constant factor is given as \(\frac{2}{3}\).

Revision Table: Key Statistical Measures

Additional Information: Properties of Normal Distribution Quartiles

The Normal Distribution is a symmetrical distribution, which means its mean, median, and mode are all equal. The spread of the distribution is determined by the standard deviation (\(\sigma\)).

• The first quartile (\(Q_1\)) and third quartile (\(Q_3\)) are equidistant from the mean (\(\mu\)).
• The distance from the mean to \(Q_1\) or \(Q_3\) is known as the quartile distance. This distance is approximately \(0.6745\sigma\).
• The interquartile range (IQR) is \(Q_3 - Q_1 = (\mu + 0.6745\sigma) - (\mu - 0.6745\sigma) = 2 \times 0.6745\sigma = 1.349\sigma\).
• The quartile deviation (QD) is half the IQR, so \(QD = \frac{1.349\sigma}{2} = 0.6745\sigma\).
• The constant \(0.6745\) comes from the standard normal distribution (mean 0, standard deviation 1). It is the value \(z\) such that the cumulative probability \(P(Z \le z)\) is 0.75. This value is often rounded or approximated, with \(\frac{2}{3}\) being a common approximation used in some contexts, especially for quick estimations or when presenting simplified relationships.