Normal Curve 10.3 Z-scores.

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Presentation transcript:

Normal Curve 10.3 Z-scores

Vocabulary: Empirical rule - Provides an estimate of the spread of the data in a normal distribution given the mean and the standard deviation. Z-scores - A statistical measurement of a scores relationship to the mean in a set of data.

Normal Curve Bell-shaped, symmetrical curve Transition points between cupping upward & downward occur at m + s and m – s As the standard deviation increases, the curve flattens & spreads As the standard deviation decreases, the curve gets taller & thinner

Can ONLY be used with normal curves! Empirical Rule Approximately 68% of the observations are within 1s of m Approximately 95% of the observations are within 2s of m Approximately 99.7% of the observations are within 3s of m Can ONLY be used with normal curves!

X-axis is standard deviations Normal Curve 34% 34% 68% 47.5% 47.5% 95% Take time to slowly click through slide. Stress that the Empirical Rule can ONLY be used with the assumption that the distribution is normal (bell-shaped curve). Sixty-eight percent of the ordered data of a normal distribution lies within one standard deviation of the mean. Ninety-five percent of the ordered data of a normal distribution lies within two standard deviations of the mean. And, 99.7% of the ordered data of a normal distribution lies within 3 standard deviations of the mean. The normal distribution here, in this example shown, has a mean of 0 and standard deviation of 1. 49.85% 49.85% 99.7% X-axis is standard deviations

The height of male students at CHS is approximately normally distributed with a mean of 71 inches and standard deviation of 2.5 inches. a) What percent of the male students are shorter than 66 inches? b) Taller than 73.5 inches? c) Between 66 & 73.5 inches? About 2.5% About 16% About 81.5%

z score Standardized score Creates the standard normal density curve Z-score normalizes samples of data so that you can compare data variability on the same “level” (m = 0 & s = 1)

What do these z scores mean? -2.3 1.8 6.1 -4.3 Note: z-score is number of deviations above or below mean 2.3 s below the mean 1.8 s above the mean 6.1 s above the mean 4.3 s below the mean

What is the z -score of a value of 27, given a set mean of 24, and a standard deviation of 2?

What is the z -score of a value of 104 What is the z -score of a value of 104.5, in a set with μ = 125 and σ = 6.2

Find the value represented by a z -score of 2 Find the value represented by a z -score of 2.403, given μ = 63 and σ = 4.25

Normalized Data Normalizing data allows you to compare different types of data For example: Duka is part of a pigmy colony and is 54 inches tall. The average female pigmy height is 50 inches with a σ of 1.5 in. Amy is 66 inches tall and the average height for her race is 60 inches with a σ of 3 in. Who is considered taller?