Normal Probability Curve

Characteristics of normal probability curve (NPC) 

• It is a unimodel curve, i.e., it has only one mode. 

• It is a bell shaped curve combining both concave and convex curves. 

• For this curve, mean, median and mode are the same. 

• The curve is perfectly symmetrical, i.e., left and right halves are the mirror images. 

• This curve is bilateral, i.e., 50% area of the curve lies to the left side and the other 50% lies to the right side. 

• The curve is asymptotic to the x-axis, i.e., it approaches but never touches the x-axis. 

• The maximum ordinate (y-coordinate) occurs at the centre and is equal to 0.3989. 

• The points of Influx (convex to concave) occur at point ±σ (±1 standard deviation unit). 

• The curve extends on both sides -3σ distance on the left to +3σ distance on the right.

Application of normal probability curve 

• It is used in behavioural researches because most of the variables of social sciences are normally distributed if a large sample or whole population is taken. 

• It is used for standardising tests. 

• Difficulty value of test items can also be compaired by normal probability curve. 

• It is also used in assigning grades. 

• Raw scores can be converted into standard scores. 

• It is also used for developing z-scores or norms of the test. 

• It is used to calculate the number of cases who achieve more or less than a given particular score. 

• It is used to calculate the number of cases who achieve within the given two limits of scores. 

• It is used to calculate the limits which include a given percentage of cases. 

• It is used to calculate the relative difficulty of test items. 

• It is used to divide a normal distribution into various categories on the basis of the ability level. 

• It is used to compare two normal distributions in teams of their mutual overlapping. 

Factors causing divergence in the normal distribution curve 

1. Selection of the sample 

• If the sample size of the individuals is small or sample is biased one, divergence is possible in the normal distribution curve. 

• Scores from small and homogeneous groups yield narrow and leptokurtic distribution. 

• Scores from small and highly heterogeneous groups yield platykurtic distribution. 

2. Unsuitable or poorly made tests 

• If the measuring tool or test is unappropriate, or poorly made, the asymmetry is possible in the distribution of scores. 

• If a test is too easy or too hard, normal distribution curve will have divergence. 

3. The trait being measured is non-normal 

• Divergence will appear when there is a real lack of normality in the traits being measured, e.g., interest, attitude, deaths in old age or early childhood due to certain degenerative diseases etc. 

4. Errors in the construction and administration of tests 

• Factors like the following during administrating the test may cause asymmetry in the distribution of scores 

  • Unclear instructions 

  • Error in timing 

  • Lack of practice 

  • Lack of motivation to complete the test. 

• Construction of unstandardised and poor item-analysed test. 

Measurement of divergence 

Measurement of skewness 

Sk = 3(M-Md)/σ
where, M = Mean
Md = median
σ = Standard Deviation 

Sk = [(P90 - P10)/2] - P50
where, P = Percentile 

Measurement of kirtosis 

Ku = Q / (P90 - P10)
where, Q = Quartile Deviation
P = Percentile