# DESCRIPTIVE STATISTICS PDF

Statistics for Engineers 4. Introduction to Statistics. Descriptive Statistics. Types of data. A variate or random variable is a quantity or attribute whose value . Examples using statistics are: Hang Seng Index, Life or car insurance rate, Descriptive Statistics - It deals with the presentation of numerical facts, or data, in. of descriptive statistics, errors in interpretations are very likely. A primary use of descriptive statistics is to determine whether the data are normally distributed.  NCSS, LLC. All Rights Reserved. Chapter Descriptive Statistics. – Summary Tables. Introduction. This procedure is used to summarize continuous data. Introduction to. Descriptive Statistics. Jackie Nicholas. Mathematics Learning Centre. University of Sydney. NSW [email protected] University of Sydney. Statistics. Descriptive Statistics gives numerical and graphic procedures to summarize a collection of data in a clear and understandable way whereas Inferential.

Positive Correlation What does it mean to say that two variables are positively correlated? If two variables are positively correlated they go up and down together their correlation coefficient is positive. Negative Correlation What does it mean to say that two variables are negatively correlated? Scores that are negatively correlated are associated in such a way that one score falls as the other rises as in the relationship between self-esteem and depression ; correlation coefficient is negative.

Uncorrelated Variables What does it mean to say that two variables are uncorrelated? If two variables are uncorrelated or not correlated the correlation coefficient is near zero.

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Look at the example from screen 6 of the exercise: Correlation Coefficient What is a correlation coefficient? The correlation coefficient is a statistic a number that indicates the strength of the association between the two variables. Why Use It? What value or benefit would a researcher gain by calculating a correlation coefficient rather than simply describing the relationship as a positive correlation or a negative correlation?

The correlation coefficient allows us to talk about the relationship with much more precision. Instead of saying simply that two variables are correlated or not correlated, we can say that the two variables have a correlation of 0.

Estimating the Relationship Look at the scatterplots and try to estimate the direction positive or negative and the strength of the relationship. Write in your guess below. Download pdf. Remember me on this computer. Descriptive Statistics 1. Bao Vang. Descriptive Statistics 1 PsychSim 5: This activity introduces you to the basic statistics that researchers use to summarize their sets of data.

The numbers below represent the scores of a group of students on a math test. Use them to perform the required calculations. Measures of Central Tendency What is a mode? The mode is the most frequent score in a distribution and is a measure of the central tendency.

What is the mode of your distribution? The median is the middle score in a distribution and is the second measure of the central tendency. What is the median of your distribution? The mean is the average of the distribution and the third measure of central tendency. How is a mean calculated? The mean is calculated by adding up all the scores and dividing the total by the number of scores. Descriptive Statistics 2 What is the mean of your distribution?

In a distribution like this, the mean is pulled toward the extreme scores at the high end of the range. Since the median is not affected by these scores, it is probably a better measure of central tendency in this case. Example 4: Draw a histogram to represent the data in the table below. Height cm Frequency 6 9 15 5 2 Solution: Histogram showing the heights Frequency The points, when joined, result usually in a smooth curve which is the form of an elongated S.

## Module 1 - Descriptive Statistics.pdf

Using the data below: Weight kg Frequency Cumulative frequency 20 — 24 5 5 25 — 29 4 9 30 — 34 3 12 35 — 39 5 17 40 — 44 5 22 45 — 49 5 27 50 — 54 18 45 55 — 59 12 57 60 — 64 3 60 The cumulative frequency or Ogive will appears as follows: Stem plots Stem — and — leaf plots A stem-and-leaf plot is used to visualize data. To set up a stem-and-leaf plot, we follow some simple steps.

First we have a set of data. Then we identify the lowest and greatest number in the data set and then we draw a vertical line. On the left hand side of the line we write the numbers that correspond to the tens and on the right hand side of the line, we will write the numbers that corresponds with the units. The digits should be arranged in order. Draw the stem plot for the following data a 13, 24, 22, 15, 33, 32, 12, 31, 26, 28, 14, 19, 20, 22, 31, The age distribution in years of a group of individual is given below.

Since in reality most measurement is not exact, hence offence is not much committed when data are rounded up. Methods of rounding are rounding to specific units. Also rounding to specific significant figures e. Again when an amount, say N This implies rounded figures have errors. For any given figure the correct figure lies in a certain range. Half of this range is the Error. Relative error Percentage Error. The length of a pole is measured as 10meters to the nearest meter. What is the range of its actual length? Calculate the percentage error. Solution The actual length of the pole will be between 9.

They are single values that give a description of the data. They are also referred to as measure of central tendency.

Some of them are arithmetic mean, geometric mean, harmonic mean, mode, and median. M The arithmetic mean average of set of observation is the sum of the observation divided by the number of observation.

Given a set of a numbers , the arithmetic mean denoted by is defined by Example 1: The ages of ten students in STS are determine the mean age. If the numbers times respectively, the for short.

## Descriptive Statistics Books

Example 2: Calculation of mean from grouped data If the items of a frequency distribution are classified in intervals, we make the assumption that every item in an interval has the mid-values of the interval and we use this midpoint for.

The table below shows the distribution of the waiting items for some customers in a certain petrol station in Abeokuta.

Waiting time in 1. Waiting in min No of customers Class mark mid-value X 1. Therefore, since. If a constant factor C is used then. The exact pension allowance paid in Nigeria to 25 workers of a company is given in the table below. Pension in N 25 30 35 40 45 No of person 7 5 6 4 3 Calculate the mean using an assumed mean Consider the data in example 3, using a suitable assume mean, compute the mean. Waiting time 1. It is always easier to select the class mark with the longest frequency as the assumed mean. It is single and easy to compute and it is the most widely used average. M The H.

M of a set of numbers is the reciprocal of the arithmetic mean of the reciprocals of the numbers. It is used when dealing with the rates of the type per such as kilometers per hour, Naira per liter. The formula is expressed thus: M If has frequency , then H. M Example: Find the harmonic mean of. M The G.

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M is an analytical method of finding the average rate of growth or decline in the values of an item over a particular period of time.

The geometric mean of a set of number is the root of the product of the number. Thus G. M If is the frequency of , then G. The rate of inflation in fire successive year in a country was. What was the average rate of inflation per year? M Average rate of inflation is Note: Relation between Arithmetic mean, Geometric and Harmonic In general, the geometric mean for a set of data is always less than or equal to the corresponding arithmetic mean but greater than or equal to the harmonic mean.

That is, H. M The equality signs hold only if all the observations are identical. The location of the median is th item. But if is even, the median is the average of the two middle observations in the ordered list. Example 1: The values of a random variable are given as Find the median. In an array: The value 0f a random variable are given as and.

Find the median. Median, Calculation of Median from a grouped data The formula for calculating the median from grouped data is defined as Where: Class size or width. The table below shows the height of 70 men randomly selected at Sango Ota. Height No of rods 8 10 14 18 9 7 4 Compute the median.

Solution Height Frequency Cumulative frequency — 8 8 — 10 18 — 14 32 — 18 50 — 9 59 — 7 66 — 4 70 The sum of first three classes frequency is 32 which therefore means that the median lies in the fourth class and this is the median class.

A set of data may have no, one, two or more modes. A distribution is said to be uni-model, bimodal and multimodal if it has one, two and more than two modes respectively. The mode of scores 2, 5, 2, 6, 7 is 2. Calculation of mode from grouped data From a grouped frequency distribution, the mode can be obtained from the formula. Mode, Where: Difference between the frequency of the modal class and the class after it.

Class size. For the table below, find the mode. Exercise 3 1. Find the mean, median and mode of the following observations: The six numbers 4, 9,8,7,4 and Y, have mean of 7. Find the value of Y. So also these are other quantity that divides a set of data in an array into different equal parts.

Such data must have been arranged in order of magnitude. Some of the partition values are: For ungrouped data, the distribution is first arranged in ascending order of magnitude. Then First Quartiles: Second Quartile: Third Quartile: DECILES The values of the variable that divide the frequency of the distribution into ten equal parts are known as deciles and are denoted by the fifth deciles is the median.

Then For a grouped data, Where Example: These include range, mean deviation, semi-interquartile range, variance, standard deviation and coefficient of variation. It is the difference between the largest and the smallest value in a set of data.

## Integrating Real-Life Data Analysis in Teaching Descriptive Statistics: A Constructivist Approach

It is commonly used in statistical quality control. However, the range may fail to discriminate if the distributions are of different types. It is good measure of spread for midrange and the quartiles. A measure of the center could be the arithmetic mean or median. Given a set of the mean deviation from the arithmetic mean is defined by: In a grouped data Example1: Below is the average of 6 heads of household randomly selected from a country. Find the i Range ii Mean iii Mean deviation from the mean iv Mean deviation from the median.

The table below shown the frequency distribution of the scores of 42 students in MTS Scores 0—9 10 — 19 20 — 29 30 — 39 40 — 49 50 — 59 60 — 69 No of student 2 5 8 12 9 5 1 Find the mean deviation from the mean for the data.

Classes midpoint 0—9 4. Thus, given as a set of observations, then the standard deviation is given by: We use when using sample instead of the population to obtain the standard deviation. The coefficients of variation are obtained by dividing the standard deviation by the mean and multiply it by Symbolically The distribution with smaller C.The median is the middle score in a distribution and is the second measure of the central tendency.