Problem 42 - Thinking Mathematically

Question

A set of data items is normally distributed with a mean of 60 and a standard deviation of 8 . In Exercises 33-48, convert each data item to a z-score. 100

Step-by-Step Solution

Answer

The z-score for 100 is 5.

Step 1: Understanding what a z-score is

A z-score is a measure of how many standard deviations a certain data point is from the mean. It is calculated by subtracting the mean from the raw score and then dividing by the standard deviation.

Step 2: Identify important values

The mean of this set of data is given as 60, the standard deviation is given as 8, and the raw score that has to be converted is 100.

Step 3: Apply the formula to find z-score

Apply the formula to calculate the z-score: z = (X - μ) / σ. Substituting the given values into the formula gives: z = (100 - 60) / 8 = 40 / 8.

Step 4: Perform the computation

Perform the division in the previous equation gives: z = 5. So, 100 is 5 standard deviations above the mean.

Key Concepts

Normal Distribution

The concept of a normal distribution is foundational in statistics and represents a continuous probability distribution that is symmetrical around its mean, indicating that data near the mean are more frequent in occurrence than data far from the mean. In other words, most of the observations cluster around the central peak and the probabilities for values further away from the mean taper off equally in both directions.

The properties of a normal distribution include its bell-shaped curve when graphed and its mean, median, and mode being equal. A perfect normal distribution would mean that exactly half of the values are to the left of the mean and the other half to the right. This distribution is very common in the real world - for instance, heights or test scores often follow a normal distribution pattern.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. It signifies how much the individual data points in a distribution deviate from the mean or average of the set. A low standard deviation indicates that the data points tend to cluster close to the mean, showing little variability, whereas a high standard deviation suggests that the data points are spread out over a wider range of values.

Mathematically, the standard deviation is the square root of the variance, where variance is the average of the squared differences from the mean. It's important for understanding the z-score since the standard deviation is used in the denominator of the z-score formula, showing how many standard deviations a particular point is from the mean.

Statistical Mean

The statistical mean, often simply called the average, is one of the most common measures of central tendency. To calculate the mean, you sum up all the values in a dataset and then divide by the number of values. The mean represents the center of the data's distribution and is used as a reference point for measuring other statistical values, such as variance and standard deviation.

In the context of the normal distribution and z-scores, the mean is key because z-scores are centered around the mean (a z-score of zero indicates a value equal to the mean). Understanding the mean is essential for interpreting z-scores and it essentially sets the 'center' of the data you're examining.

Data Normalization

Data normalization is a process used to standardize the range of independent variables or features of data. In the context of z-scores, normalization refers to the conversion of individual data points to a common scale without distorting differences in the ranges of values.

For example, by converting each data item to a z-score, we can compare different data points within the same dataset or even across different datasets on a standardized scale. Another advantage of data normalization through z-scores is that it allows us to calculate probabilities and relate the data to a standard normal distribution so we can make statistical inferences about the data under consideration.